Question

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

Answer: The probability that one or the other or both will wait 5 minutes or longer at the cashier's counter is 0.36.

Step-by-step solution

Complementary Probability for the man

The probability of waiting 5 minutes or longer for checkout is 0.2. To find the probability that the man will wait less than 5 minutes, we need to find the complement of this event, which is the probability of not waiting 5 minutes or longer. To find the complementary probability, we subtract the given probability from 1: Man waiting less than 5 minutes = 1 - Probability of waiting 5 minutes or more = 1 - 0.2 = 0.8

Probability of both waiting less than 5 minutes

Both the man and his wife's checkout times are independent events. Therefore, to find the probability of both of them waiting less than 5 minutes, we can multiply their individual probabilities: P(Both waiting less than 5 minutes) = P(Man waiting less than 5 minutes) × P(Wife waiting less than 5 minutes) = 0.8 × 0.8 = 0.64

Probability of one or the other or both waiting 5 minutes or longer

To find the probability of one or the other or both waiting 5 minutes or longer, we can first find the probability of both waiting less than 5 minutes, which we already found in Step 2. Then, we use the complementary probability to find the answer to this part: P(One or the other or both waiting 5 minutes or longer) = 1 - P(Both waiting less than 5 minutes) = 1 - 0.64 = 0.36 To summarize the results: a. The probability that the man will wait less than 5 minutes is 0.8 b. The probability that both the man and his wife will be checked out in less than 5 minutes is 0.64 c. The probability that one or the other or both will wait 5 minutes or longer is 0.36

Key concepts

Complementary Probability

In probability, the concept of complementary events is a powerful tool that simplifies calculations. The complement of an event is everything that is not the event itself. Hence, to find the complementary probability, you subtract the probability of the event from 1. For example, in our supermarket scenario, the probability that the man will wait five minutes or longer is 0.2. Therefore, the probability of the complement event, which is the man waiting less than five minutes, can be calculated by subtracting 0.2 from 1, giving us 0.8. This means there is an 80% chance he will be checked out in under five minutes.
Some key points about complementary probabilities:

• The sum of the probabilities of an event and its complement is always equal to 1.
• Complementary probability is useful in scenarios where it is easier to calculate the probability of an event not occurring.

Independent Events

Understanding independent events is crucial in probability. Two events are considered independent if the outcome of one event does not influence the outcome of the other. This property is pivotal when dealing with multiple random events.In the example of the man and his wife shopping separately, their checkout times are independent events. This means that the probability of one person being checked out in less than five minutes is independent of the other person's checkout time.
When events are independent:

• The occurrence of one event provides no information about the occurrence of the other event.
• The probability of both events occurring is the product of their individual probabilities.

To illustrate this, the probability that both the man and his wife are checked out in less than five minutes is calculated by multiplying their individual probabilities: \( P(\text{Both less than 5 minutes}) = P(\text{Man less than 5}) \times P(\text{Wife less than 5}) = 0.8 \times 0.8 = 0.64 \) This result, 0.64, shows that there is a 64% chance both individuals will finalize checkout in under five minutes.

Multiplication Rule for Probabilities

The multiplication rule for probabilities is essential when working with scenarios involving several events. This rule simplifies the calculation of the probability of multiple independent events occurring together. The multiplication rule states that for independent events \( A \) and \( B \), the probability of both events occurring, \( P(A \text{ and } B) \), is the product of their separate probabilities: \[ P(A \cap B) = P(A) \times P(B) \]Applying this rule to our problem, where both the man and his wife check out at separate counters, we calculate the joint probability of both waiting less than five minutes by multiplying each individual's probability of waiting less than five minutes, which is 0.8 for both. Thus, \( P(\text{Both less than 5 minutes}) = 0.8 \times 0.8 = 0.64 \)Key takeaways from the multiplication rule for probabilities include:

• The multiplication rule is only applicable to independent events.
• It helps in calculating the simultaneous occurrence of two or more events.
• Real-world examples often involve independent events, such as flipping a coin or rolling a die.

Overall, understanding the multiplication rule aids in solving complex problems involving multiple steps or layers of probability.