91Ó°ÊÓ

Many fire stations handle emergency calls for medical assistance as well as calls requesting firefighting equipment. A particular station says that the probability that an incoming call is for medical assistance is .85. This can be expressed as \(P(\) call is for medical assistance \()=.85\). a. Give a relative frequency interpretation of the given probability. b. What is the probability that a call is not for medical assistance? c. Assuming that successive calls are independent of one another, calculate the probability that two successive calls will both be for medical assistance. d. Still assuming independence, calculate the probability that for two successive calls, the first is for medical assistance and the second is not for medical assistance. e. Still assuming independence, calculate the probability that exactly one of the next two calls will be for medical assistance. (Hint: There are two different possibilities. The one call for medical assistance might be the first call, or it might be the second call.) f. Do you think that it is reasonable to assume that. the requests made in successive calls are independent? Explain.

Step by step solution

01

Interpretation of Probability

Since the probability that an incoming call is for medical assistance is .85, it means that out of every 100 calls, about 85 would be expected to be for medical assistance.

02

Probability of a Call Not for Medical Assistance

The probability that an event does not occur is 1 minus the probability that it does occur. So, the probability that a call is not for medical assistance is \(1-0.85=0.15\)

03

Probability of Successive Calls for Medical Assistance

Assuming that successive calls are independent of one another, the probability that both will be for medical assistance is calculated by multiplying the probabilities together, i.e. \(0.85 * 0.85 = 0.7225\)

04

Probability of First Call for Medical Assistance and Second Not

The probability that the first call is for medical assistance and the second is not is calculated by multiplying the probabilities of the respective events together, i.e. \(0.85 * 0.15 = 0.1275\)

05

Probability of Exactly One Call for Medical Assistance

Since exactly one of the two calls is for medical assistance, there are two ways this can occur: The first call is for medical assistance and the second is not, or the first call is not for medical assistance and the second is. Each scenario has a probability of 0.1275. Adding these gives a total probability of \(0.1275 + 0.1275 = 0.255\)

06

Analysis of Independence Assumption

It may not be reasonable to assume that requests made in successive calls are independent. In a real-world scenario, there may be factors which influence the type of requests made in successive calls. For example, if there's an accident causing multiple injuries, several successive calls could be for medical assistance.

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Key Concepts

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

Relative Frequency

Relative frequency is a way of interpreting probability based on the long-term observation of events. In our exercise with the fire station, the probability that an incoming call is for medical assistance is given as 0.85. This probability can be understood through relative frequency by saying that, if you were to observe many calls, approximately 85 out of every 100 calls will be for medical assistance.

It's important to understand that relative frequency is derived from actual data over a period of time. So, if this pattern holds consistently, we can use this information to predict future occurrences.

This interpretation is particularly useful in real-world scenarios where patterns emerge from experiences over time.

Independent Events

In probability, events are considered independent if the occurrence of one event does not affect the probability of another. In this context, when the problem assumes that successive calls to the fire station are independent, it means that the nature of one call (whether it is for medical assistance or not) does not change the likelihood of the next call’s nature.

For example, calculating the probability of both calls being for medical assistance involves multiplying the probabilities: 0.85 (first call is for medical) × 0.85 (second call is for medical), resulting in 0.7225. The independence assumption simplifies this calculation.

• If successive calls influence each other, such as during a large-scale event, this assumption may not hold.
• Understanding independence helps in precise probability calculations, especially in sequences of events.

Complement Rule

The complement rule helps us find the probability of an event not occurring. The essence of this rule is that the probability of an event not occurring is 1 minus the probability of that event occurring.

In the exercise, the probability that a call is for medical assistance is 0.85. Therefore, using the complement rule, the probability that a call is not for medical assistance is calculated as 1 - 0.85 = 0.15.

This rule is fundamental in probability as it provides an easy way to calculate probabilities of complementary events, which are often easier to determine in many practical scenarios.

Joint Probability

Joint probability refers to the probability of two or more events happening simultaneously. In our problem involving the fire station, we calculate joint probabilities under the assumption of independence.

For instance, to find the probability that the first call is for medical assistance and the second is not, we multiply their individual probabilities: 0.85 (first call) × 0.15 (second call not for medical) = 0.1275.

Each potential sequence of calls is considered separately, and their probabilities are added to solve for specific scenarios, such as exactly one call being for medical assistance. Hence, joint probability helps us understand and predict the combination of events.

• Consider all possible outcomes to ensure accurate probability computation.
• Joint probability requires careful attention to whether events are independent or not.