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Chapter 4: Problem 125

Suppose that at a particular supermarket the probability of waiting 5 minutes or longer for checkout at the cashier's counter is .2. On a given day, a man and his wife decide to shop individually at the market, each checking out at different cashier counters. They both reach cashier counters at the same time. a. What is the probability that the man will wait less than 5 minutes for checkout? b. What is probability that both the man and his wife will be checked out in less than 5 minutes? (Assume that the checkout times for the two are independent events.) c. What is the probability that one or the other or both will wait 5 minutes or longer?

Short Answer

Expert verified
Answer: The probability that one or both will wait 5 minutes or longer for checkout is 0.36.

Step by step solution

01

a. The probability that the man will wait less than 5 minutes for checkout

Since waiting 5 minutes or longer (M) and waiting less than 5 minutes (M') are complementary events, the probability of M' is given by: P(M') = 1 - P(M) = 1 - 0.2 Calculate the result: P(M') = 0.8 So, the probability that the man will wait less than 5 minutes for checkout is 0.8.
02

b. The probability that both the man and his wife will wait less than 5 minutes

Because the man's and his wife's checkout times are independent events, the probability that both will wait less than 5 minutes is the product of their respective probabilities: P(M' and W') = P(M') * P(W') We know P(M') = 0.8 and P(W') = 1 - P(W) = 1 - 0.2 = 0.8. So: P(M' and W') = 0.8 * 0.8 Compute the result: P(M' and W') = 0.64 Thus, the probability that both the man and his wife will wait less than 5 minutes is 0.64.
03

c. The probability that one or both will wait 5 minutes or longer

We can find this probability using the complement: the probability that neither of them waits 5 minutes or longer. This is the same as the probability that both wait less than 5 minutes, which we've already found: P(not (M or W)) = P(M' and W') = 0.64 Now, we can apply the complement rule: P(M or W) = 1 - P(not (M or W)) P(M or W) = 1 - 0.64 Compute the result: P(M or W) = 0.36 Therefore, the probability that one or both will wait 5 minutes or longer is 0.36.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability, independent events refer to two or more events that do not affect each other. This means that the outcome of one event does not change the probability of the other event occurring. In the exercise, we see this concept demonstrated with the shopping scenario.
  • The man's checkout time is considered independent of his wife's checkout time.
  • This independence allows us to calculate the combined probability by multiplying the individual probabilities.
So, if both have a probability of 0.8 for waiting less than 5 minutes, we multiply these two probabilities: 0.8 * 0.8 = 0.64. This calculation gives us the probability that both will be checked out in less than 5 minutes. Remember, in real life, events might not always be exactly independent, but this assumption simplifies calculations for scenarios like this exercise.
Complementary Events
Complementary events are pairs of outcomes that cover all possibilities. When one event occurs, the other does not. For these events, the sum of their probabilities always equals 1. In our shopping example:
  • Waiting 5 minutes or longer is one event, and waiting less than 5 minutes is its complement.
  • The probability of waiting longer is given as 0.2. So, the complementary probability that the man waits less than 5 minutes is 1 - 0.2 = 0.8.
Understanding complementary events lets us easily find the probability of one event if we know its complement. This principle is used in various steps of the solution, particularly when determining the likelihood of checking out quickly versus experiencing a delay.
Waiting Time
Waiting time in probability refers to the duration until an event happens, such as the time spent at a checkout line.
  • Probabilities can describe segments of waiting time, e.g., less than or more than 5 minutes.
  • In our problem, we consider checkout times separately for the man and the wife, with each described probabilistically.
By examining different waiting scenarios, we get insights into average experiences at the checkout, like how likely customers might encounter quick or slower services. Understanding waiting times with probabilities can help with managing expectations and optimizing processes in real-world situations, like at supermarkets.

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