Find The Probability Of Exactly Three Successes In Eight Trials Of A Binomial Experiment In Which The Probability Of Success Is 45 % 45\% 45% .Calculate: P ( 3 ) = 56 × 0.0911 × ( 0.55 ) 8 − 3 P(3) = 56 \times 0.0911 \times (0.55)^{8-3} P ( 3 ) = 56 × 0.0911 × ( 0.55 ) 8 − 3 Solve Part Of The

Introduction

In probability theory, the binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this article, we will explore how to find the probability of exactly three successes in eight trials of a binomial experiment, where the probability of success is $45\%$. We will use the formula for the binomial probability and calculate the result step by step.

The Binomial Probability Formula

The binomial probability formula is given by:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $P(X = k)$ is the probability of exactly $k$ successes in $n$ trials
• $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time, also written as $C(n, k)$ or $nCk$
• $p$ is the probability of success in a single trial
• $n$ is the number of trials
• $k$ is the number of successes

Calculating the Probability of Exactly Three Successes

In this problem, we have:

• $n = 8$ (number of trials)
• $k = 3$ (number of successes)
• $p = 0.45$ (probability of success)

We need to calculate the probability of exactly three successes in eight trials. Using the binomial probability formula, we get:

$$P(3) = \binom{8}{3} (0.45)^3 (1-0.45)^{8-3}$$

Calculating the Number of Combinations

The number of combinations of $n$ items taken $k$ at a time is given by:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

where $n!$ is the factorial of $n$. In this case, we have:

$$\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}$$

Calculating the factorials, we get:

$$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$$

$$3! = 3 \times 2 \times 1 = 6$$

$$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$

Substituting these values, we get:

$$\binom{8}{3} = \frac{40320}{6 \times 120} = \frac{40320}{720} = 56$$

Calculating the Probability

Now that we have the number of combinations, we can calculate the probability of exactly three successes in eight trials:

$$P(3) = 56 \times (0.45)^3 \times (1-0.45)^{8-3}$$

$$P(3) = 56 \times (0.45)^3 \times (0.55)^5$$

Calculating the Powers

To calculate the powers, we need to raise the numbers to the specified powers:

$$(0.45)^3 = 0.45 \times 0.45 \times 0.45 = 0.0911$$

$$(0.55)^5 = 0.55 \times 0.55 \times 0.55 \times 0.55 \times 0.55 = 0.000244140625$$

However, we can simplify this calculation by using the fact that $(0.55)^5 = (0.55)^{8-3}$. Therefore, we can write:

$$(0.55)^5 = (0.55)^{8-3} = (0.55)^8 \times (0.55)^{-3}$$

Using the property of exponents that $a^{-n} = \frac{1}{a^n}$, we can rewrite this as:

$$(0.55)^5 = (0.55)^8 \times \frac{1}{(0.55)^3}$$

Now, we can simplify the calculation by canceling out the $(0.55)^3$ terms:

$$(0.55)^5 = (0.55)^8 \times \frac{1}{(0.55)^3} = (0.55)^5$$

Therefore, we can simplify the calculation of the probability as:

$$P(3) = 56 \times 0.0911 \times (0.55)^5$$

Simplifying the Calculation

To simplify the calculation, we can use the fact that $(0.55)^5 = (0.55)^{8-3}$. Therefore, we can write:

$$(0.55)^5 = (0.55)^{8-3} = (0.55)^8 \times (0.55)^{-3}$$

Using the property of exponents that $a^{-n} = \frac{1}{a^n}$, we can rewrite this as:

$$(0.55)^5 = (0.55)^8 \times \frac{1}{(0.55)^3}$$

Now, we can simplify the calculation by canceling out the $(0.55)^3$ terms:

$$(0.55)^5 = (0.55)^8 \times \frac{1}{(0.55)^3} = (0.55)^5$$

Therefore, we can simplify the calculation of the probability as:

$$P(3) = 56 \times 0.0911 \times (0.55)^5$$

Calculating the Final Result

Now that we have simplified the calculation, we can calculate the final result:

$$P(3) = 56 \times 0.0911 \times (0.55)^5$$

$$P(3) = 56 \times 0.0911 \times 0.000244140625$$

$$P(3) = 0.0013$$

Therefore, the probability of exactly three successes in eight trials is $0.0013$ or $0.13\%$.

Conclusion

$$$$$$

Q: What is the binomial probability formula?

A: The binomial probability formula is given by:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $P(X = k)$ is the probability of exactly $k$ successes in $n$ trials
• $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time
• $p$ is the probability of success in a single trial
• $n$ is the number of trials
• $k$ is the number of successes

Q: What is the number of combinations formula?

A: The number of combinations formula is given by:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

where $n!$ is the factorial of $n$.

Q: How do I calculate the probability of exactly three successes in eight trials?

A: To calculate the probability of exactly three successes in eight trials, you can use the binomial probability formula:

$$P(3) = \binom{8}{3} (0.45)^3 (1-0.45)^{8-3}$$

First, calculate the number of combinations:

$$\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}$$

Then, calculate the powers:

$$(0.45)^3 = 0.45 \times 0.45 \times 0.45 = 0.0911$$

$$(1-0.45)^{8-3} = (0.55)^5 = 0.000244140625$$

Finally, multiply the results together:

$$P(3) = 56 \times 0.0911 \times 0.000244140625$$

Q: What is the probability of exactly three successes in eight trials?

A: The probability of exactly three successes in eight trials is $0.0013$ or $0.13\%$.

Q: How do I use the binomial probability formula in real-life situations?

A: The binomial probability formula can be used in various real-life situations, such as:

• Predicting the number of successes in a series of trials
• Calculating the probability of a certain outcome in a binomial experiment
• Modeling the behavior of a population or a system

For example, a company may want to know the probability of selling exactly 10 units of a new product in a marketing campaign. Using the binomial probability formula, the company can calculate the probability of selling exactly 10 units based on the number of customers, the probability of a customer buying the product, and the number of trials.

Q: What are some common applications of the binomial probability formula?

A: Some common applications of the binomial probability formula include:

• Quality control: Calculating the probability of a certain number of defects in a batch of products
• Marketing: Predicting the number of sales or conversions in a marketing campaign
• Finance: Calculating the probability of a certain return on investment or a certain level of risk
• Medicine: Calculating the probability of a certain outcome in a medical trial or a certain level of risk associated with a treatment

Q: What are some common mistakes to avoid when using the binomial probability formula?

A: Some common mistakes to avoid when using the binomial probability formula include:

• Not understanding the assumptions of the binomial distribution (e.g., independence of trials, constant probability of success)
• Not calculating the number of combinations correctly
• Not using the correct values for the parameters (e.g., number of trials, probability of success)
• Not interpreting the results correctly

By understanding the binomial probability formula and its applications, you can make informed decisions in various fields and avoid common mistakes.