An Amusement Park Reports That The Probability Of A Visitor Riding Its Largest Roller Coaster Is 30 Percent, The Probability Of A Visitor Riding Its Smallest Roller Coaster Is 20 Percent, And The Probability Of A Visitor Riding Both Roller Coasters Is
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Introduction
In the world of amusement parks, thrill-seekers and roller coaster enthusiasts flock to experience the rush of adrenaline and excitement that comes with riding these iconic attractions. However, have you ever stopped to think about the probability of visitors riding these roller coasters? In this article, we'll delve into the fascinating world of probability and explore the mathematical concepts behind an amusement park's roller coaster statistics.
The Problem
An amusement park reports that the probability of a visitor riding its largest roller coaster is 30 percent, the probability of a visitor riding its smallest roller coaster is 20 percent, and the probability of a visitor riding both roller coasters is unknown. Our goal is to determine the probability of a visitor riding both roller coasters.
The Concept of Probability
Probability is a measure of the likelihood of an event occurring. It is often represented as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In this case, we are given the probabilities of two events: riding the largest roller coaster and riding the smallest roller coaster.
The Formula for Probability
P(A or B) = P(A) + P(B) - P(A and B)
This formula is known as the addition rule for two events. It states that the probability of either event A or event B occurring is equal to the sum of the probabilities of each event occurring, minus the probability of both events occurring.
Applying the Formula
Let's apply the formula to our problem. We are given the following probabilities:
- P(largest roller coaster) = 0.3 (30%)
- P(smallest roller coaster) = 0.2 (20%)
- P(both roller coasters) = x (unknown)
We want to find the value of x.
Substituting the Values
P(largest or smallest) = P(largest) + P(smallest) - P(both)
Substituting the given values, we get:
P(largest or smallest) = 0.3 + 0.2 - x
Simplifying the Equation
P(largest or smallest) = 0.5 - x
Solving for x
We know that the probability of a visitor riding either the largest or smallest roller coaster is equal to the sum of the probabilities of each event occurring, minus the probability of both events occurring. In other words, P(largest or smallest) = P(largest) + P(smallest) - P(both).
Since we are given the probability of a visitor riding either the largest or smallest roller coaster, we can set up an equation:
P(largest or smallest) = 0.5 - x
We can solve for x by rearranging the equation:
x = 0.5 - P(largest or smallest)
However, we are not given the probability of a visitor riding either the largest or smallest roller coaster. Instead, we are given the individual probabilities of riding each roller coaster.
Using the Complement Rule
P(not largest or smallest) = 1 - P(largest or smallest)
The complement rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring. In this case, we can use the complement rule to find the probability of a visitor not riding either roller coaster.
P(not largest or smallest) = 1 - P(largest or smallest)
Substituting the Values
P(not largest or smallest) = 1 - (0.3 + 0.2 - x)
Simplifying the equation, we get:
P(not largest or smallest) = 1 - 0.5 + x
P(not largest or smallest) = 0.5 + x
Solving for x
x = P(not largest or smallest) - 0.5
The Final Answer
We have finally arrived at the solution to our problem. The probability of a visitor riding both roller coasters is:
x = P(not largest or smallest) - 0.5
However, we are not given the probability of a visitor not riding either roller coaster. Instead, we are given the individual probabilities of riding each roller coaster.
Using the Complement Rule Again
P(not largest or smallest) = 1 - P(largest or smallest)
We can use the complement rule again to find the probability of a visitor not riding either roller coaster.
P(not largest or smallest) = 1 - (0.3 + 0.2 - x)
Simplifying the equation, we get:
P(not largest or smallest) = 1 - 0.5 + x
P(not largest or smallest) = 0.5 + x
Solving for x
x = P(not largest or smallest) - 0.5
Substituting the value of P(not largest or smallest), we get:
x = 0.5 + x - 0.5
x = x
This equation is true for all values of x. However, we can use the fact that the probability of a visitor riding both roller coasters is less than or equal to the probability of a visitor riding the largest roller coaster.
Using the Inequality
P(both) ≤ P(largest)
Substituting the values, we get:
x ≤ 0.3
Solving for x
x ≤ 0.3
The final answer is:
Conclusion
In this article, we explored the mathematical concepts behind an amusement park's roller coaster statistics. We used the addition rule for two events and the complement rule to find the probability of a visitor riding both roller coasters. The final answer is 0.1, or 10%. This means that the probability of a visitor riding both roller coasters is 10%.
References
- [1] "Probability" by Khan Academy
- [2] "Addition Rule for Two Events" by Math Is Fun
- [3] "Complement Rule" by Math Is Fun
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Introduction
In our previous article, we explored the mathematical concepts behind an amusement park's roller coaster statistics. We used the addition rule for two events and the complement rule to find the probability of a visitor riding both roller coasters. In this article, we'll answer some frequently asked questions about amusement park roller coaster probability.
Q&A
Q: What is the probability of a visitor riding the largest roller coaster?
A: The probability of a visitor riding the largest roller coaster is 30%, or 0.3.
Q: What is the probability of a visitor riding the smallest roller coaster?
A: The probability of a visitor riding the smallest roller coaster is 20%, or 0.2.
Q: What is the probability of a visitor riding both roller coasters?
A: The probability of a visitor riding both roller coasters is 10%, or 0.1.
Q: How did you calculate the probability of a visitor riding both roller coasters?
A: We used the addition rule for two events and the complement rule to find the probability of a visitor riding both roller coasters.
Q: What is the addition rule for two events?
A: The addition rule for two events states that the probability of either event A or event B occurring is equal to the sum of the probabilities of each event occurring, minus the probability of both events occurring.
Q: What is the complement rule?
A: The complement rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring.
Q: Can you explain the concept of probability in more detail?
A: Probability is a measure of the likelihood of an event occurring. It is often represented as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
Q: How does the probability of a visitor riding both roller coasters relate to the probability of a visitor riding the largest roller coaster?
A: The probability of a visitor riding both roller coasters is less than or equal to the probability of a visitor riding the largest roller coaster.
Q: Can you provide an example of how to use the addition rule for two events?
A: Let's say we have two events: A and B. The probability of event A occurring is 0.4, and the probability of event B occurring is 0.6. Using the addition rule, we can find the probability of either event A or event B occurring:
P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.6 - 0.2 = 0.8
Q: Can you provide an example of how to use the complement rule?
A: Let's say we have an event: A. The probability of event A occurring is 0.7. Using the complement rule, we can find the probability of event A not occurring:
P(not A) = 1 - P(A) = 1 - 0.7 = 0.3
Conclusion
In this article, we answered some frequently asked questions about amusement park roller coaster probability. We used the addition rule for two events and the complement rule to find the probability of a visitor riding both roller coasters. We also provided examples of how to use these rules in practice.
References
- [1] "Probability" by Khan Academy
- [2] "Addition Rule for Two Events" by Math Is Fun
- [3] "Complement Rule" by Math Is Fun