Quiz Is Accepting Submissions Until Monday, March 10, 2025, At 2:35 PM.Question 13 (1 Point)A Company Ships Computer Components In Boxes That Contain 90 Items. Assume That The Probability Of A Defective Item Is P P P . Use The Geometric Mean To

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Introduction to the Geometric Mean

The geometric mean is a mathematical concept used to calculate the average of a set of numbers, particularly when the numbers are in a multiplicative relationship. In probability theory, the geometric mean is used to calculate the expected value of a random variable that follows a geometric distribution. In this article, we will explore how to use the geometric mean to calculate the probability of a defective item in a shipment of computer components.

The Problem

A company ships computer components in boxes that contain 90 items. Assume that the probability of a defective item is pp. We want to use the geometric mean to calculate the probability of at least one defective item in a box.

The Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of trials until the first success, where the probability of success is pp. In this case, the probability of a defective item is pp, and we want to calculate the probability of at least one defective item in a box.

Calculating the Probability of at Least One Defective Item

Let XX be the number of defective items in a box. We want to calculate the probability of Xβ‰₯1X \geq 1. Using the geometric distribution, we can write:

P(Xβ‰₯1)=1βˆ’P(X=0)P(X \geq 1) = 1 - P(X = 0)

where P(X=0)P(X = 0) is the probability of no defective items in a box.

Using the Geometric Mean

The geometric mean of a set of numbers is defined as the nth root of the product of the numbers, where n is the number of numbers. In this case, we have 90 items in a box, and we want to calculate the probability of at least one defective item. We can use the geometric mean to calculate the probability of no defective items in a box:

P(X=0)=(1βˆ’p)90P(X = 0) = (1-p)^{90}

where pp is the probability of a defective item.

Calculating the Probability of at Least One Defective Item Using the Geometric Mean

Now, we can use the geometric mean to calculate the probability of at least one defective item in a box:

P(Xβ‰₯1)=1βˆ’(1βˆ’p)90P(X \geq 1) = 1 - (1-p)^{90}

This is the probability of at least one defective item in a box, using the geometric mean.

Example

Suppose the probability of a defective item is p=0.01p = 0.01. We can calculate the probability of at least one defective item in a box using the geometric mean:

P(Xβ‰₯1)=1βˆ’(1βˆ’0.01)90P(X \geq 1) = 1 - (1-0.01)^{90}

P(Xβ‰₯1)=1βˆ’(0.99)90P(X \geq 1) = 1 - (0.99)^{90}

Q&A: Geometric Mean in Probability Calculations

Q: What is the geometric mean?

A: The geometric mean is a mathematical concept used to calculate the average of a set of numbers, particularly when the numbers are in a multiplicative relationship.

Q: How is the geometric mean used in probability calculations?

A: In probability theory, the geometric mean is used to calculate the expected value of a random variable that follows a geometric distribution. The geometric distribution models the number of trials until the first success, where the probability of success is pp.

Q: What is the geometric distribution?

A: The geometric distribution is a discrete probability distribution that models the number of trials until the first success, where the probability of success is pp.

Q: How is the geometric mean used to calculate the probability of at least one defective item in a box?

A: Let XX be the number of defective items in a box. We want to calculate the probability of Xβ‰₯1X \geq 1. Using the geometric distribution, we can write:

P(Xβ‰₯1)=1βˆ’P(X=0)</span></p><p>where<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/xβˆ’tex">P(X=0)</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="marginβˆ’right:0.07847em;">X</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">0</span><spanclass="mclose">)</span></span></span></span>istheprobabilityofnodefectiveitemsinabox.</p><h2><strong>Q:Howistheprobabilityofnodefectiveitemsinaboxcalculatedusingthegeometricmean?</strong></h2><p>A:Theprobabilityofnodefectiveitemsinaboxiscalculatedusingthegeometricmeanasfollows:</p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mostretchy="false">)</mo><mo>=</mo><mostretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mi>p</mi><msup><mostretchy="false">)</mo><mn>90</mn></msup></mrow><annotationencoding="application/xβˆ’tex">P(X=0)=(1βˆ’p)90</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="marginβˆ’right:0.07847em;">X</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">0</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal">p</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>where<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotationencoding="application/xβˆ’tex">p</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;verticalβˆ’align:βˆ’0.1944em;"></span><spanclass="mordmathnormal">p</span></span></span></span>istheprobabilityofadefectiveitem.</p><h2><strong>Q:Whatistheformulaforcalculatingtheprobabilityofatleastonedefectiveiteminaboxusingthegeometricmean?</strong></h2><p>A:Theformulaforcalculatingtheprobabilityofatleastonedefectiveiteminaboxusingthegeometricmeanis:</p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mostretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mi>p</mi><msup><mostretchy="false">)</mo><mn>90</mn></msup></mrow><annotationencoding="application/xβˆ’tex">P(Xβ‰₯1)=1βˆ’(1βˆ’p)90</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="marginβˆ’right:0.07847em;">X</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">β‰₯</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;verticalβˆ’align:βˆ’0.0833em;"></span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal">p</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p><h2><strong>Q:Canyouprovideanexampleofhowtousethegeometricmeantocalculatetheprobabilityofatleastonedefectiveiteminabox?</strong></h2><p>A:Supposetheprobabilityofadefectiveitemis<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0.01</mn></mrow><annotationencoding="application/xβˆ’tex">p=0.01</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;verticalβˆ’align:βˆ’0.1944em;"></span><spanclass="mordmathnormal">p</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">0.01</span></span></span></span>.Wecancalculatetheprobabilityofatleastonedefectiveiteminaboxusingthegeometricmeanasfollows:</p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mostretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mn>0.01</mn><msup><mostretchy="false">)</mo><mn>90</mn></msup></mrow><annotationencoding="application/xβˆ’tex">P(Xβ‰₯1)=1βˆ’(1βˆ’0.01)90</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="marginβˆ’right:0.07847em;">X</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">β‰₯</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;verticalβˆ’align:βˆ’0.0833em;"></span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">0.01</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mostretchy="false">(</mo><mn>0.99</mn><msup><mostretchy="false">)</mo><mn>90</mn></msup></mrow><annotationencoding="application/xβˆ’tex">P(Xβ‰₯1)=1βˆ’(0.99)90</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="marginβˆ’right:0.07847em;">X</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">β‰₯</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;verticalβˆ’align:βˆ’0.0833em;"></span><spanclass="mord">1</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">βˆ’</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">0.99</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p><h2><strong>Q:Whatisthesignificanceofthegeometricmeaninprobabilitycalculations?</strong></h2><p>A:Thegeometricmeanisapowerfultoolinprobabilitycalculations,particularlywhendealingwithgeometricdistributions.Itallowsustocalculatetheexpectedvalueofarandomvariableandtheprobabilityofatleastonedefectiveiteminabox.</p><h2><strong>Q:Canthegeometricmeanbeusedinotherareasofprobabilitycalculations?</strong></h2><p>A:Yes,thegeometricmeancanbeusedinotherareasofprobabilitycalculations,suchascalculatingtheprobabilityofatleastonesuccessinaseriesofindependenttrials.</p><h2><strong>Q:Whataresomecommonapplicationsofthegeometricmeaninprobabilitycalculations?</strong></h2><p>A:Somecommonapplicationsofthegeometricmeaninprobabilitycalculationsinclude:</p><ul><li>Calculatingtheprobabilityofatleastonedefectiveiteminabox</li><li>Calculatingtheprobabilityofatleastonesuccessinaseriesofindependenttrials</li><li>Calculatingtheexpectedvalueofarandomvariablethatfollowsageometricdistribution</li></ul><h2><strong>Q:Howcanthegeometricmeanbeusedinrealβˆ’worldscenarios?</strong></h2><p>A:Thegeometricmeancanbeusedinrealβˆ’worldscenariossuchas:</p><ul><li>Qualitycontrol:Calculatingtheprobabilityofatleastonedefectiveiteminaboxtoensurequalitycontrol</li><li>Insurance:Calculatingtheprobabilityofatleastoneclaiminaseriesofindependenttrialstodetermineinsurancepremiums</li><li>Finance:Calculatingtheexpectedvalueofarandomvariablethatfollowsageometricdistributiontodetermineinvestmentreturns.</li></ul>P(X \geq 1) = 1 - P(X = 0) </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X = 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> is the probability of no defective items in a box.</p> <h2><strong>Q: How is the probability of no defective items in a box calculated using the geometric mean?</strong></h2> <p>A: The probability of no defective items in a box is calculated using the geometric mean as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>90</mn></msup></mrow><annotation encoding="application/x-tex">P(X = 0) = (1-p)^{90} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> is the probability of a defective item.</p> <h2><strong>Q: What is the formula for calculating the probability of at least one defective item in a box using the geometric mean?</strong></h2> <p>A: The formula for calculating the probability of at least one defective item in a box using the geometric mean is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mo stretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>90</mn></msup></mrow><annotation encoding="application/x-tex">P(X \geq 1) = 1 - (1-p)^{90} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">β‰₯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: Can you provide an example of how to use the geometric mean to calculate the probability of at least one defective item in a box?</strong></h2> <p>A: Suppose the probability of a defective item is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0.01</mn></mrow><annotation encoding="application/x-tex">p = 0.01</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.01</span></span></span></span>. We can calculate the probability of at least one defective item in a box using the geometric mean as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mo stretchy="false">(</mo><mn>1</mn><mo>βˆ’</mo><mn>0.01</mn><msup><mo stretchy="false">)</mo><mn>90</mn></msup></mrow><annotation encoding="application/x-tex">P(X \geq 1) = 1 - (1-0.01)^{90} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">β‰₯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">0.01</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>β‰₯</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>βˆ’</mo><mo stretchy="false">(</mo><mn>0.99</mn><msup><mo stretchy="false">)</mo><mn>90</mn></msup></mrow><annotation encoding="application/x-tex">P(X \geq 1) = 1 - (0.99)^{90} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">β‰₯</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">βˆ’</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0.99</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">90</span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: What is the significance of the geometric mean in probability calculations?</strong></h2> <p>A: The geometric mean is a powerful tool in probability calculations, particularly when dealing with geometric distributions. It allows us to calculate the expected value of a random variable and the probability of at least one defective item in a box.</p> <h2><strong>Q: Can the geometric mean be used in other areas of probability calculations?</strong></h2> <p>A: Yes, the geometric mean can be used in other areas of probability calculations, such as calculating the probability of at least one success in a series of independent trials.</p> <h2><strong>Q: What are some common applications of the geometric mean in probability calculations?</strong></h2> <p>A: Some common applications of the geometric mean in probability calculations include:</p> <ul> <li>Calculating the probability of at least one defective item in a box</li> <li>Calculating the probability of at least one success in a series of independent trials</li> <li>Calculating the expected value of a random variable that follows a geometric distribution</li> </ul> <h2><strong>Q: How can the geometric mean be used in real-world scenarios?</strong></h2> <p>A: The geometric mean can be used in real-world scenarios such as:</p> <ul> <li>Quality control: Calculating the probability of at least one defective item in a box to ensure quality control</li> <li>Insurance: Calculating the probability of at least one claim in a series of independent trials to determine insurance premiums</li> <li>Finance: Calculating the expected value of a random variable that follows a geometric distribution to determine investment returns.</li> </ul>