Question

question 4 of 5
many fire stations handle more emergency calls for medical help than for fires. at one fire station, 81% of incoming calls are for medical help. suppose we choose 4 incoming calls to the station at random.
(a) find the probability that all 4 calls are for medical help.
(round to 4 decimal places. leave your answer in decimal form.)
(b) whats the probability that at least 1 of the calls is not for medical help?
(round to 4 decimal places. leave your answer in decimal form.)
(c) explain why the calculation in part (a) may not be valid if we choose 4 consecutive calls to the station.
the calculation in part (a) might not be valid because the 4 consecutive calls being medical may not be
events.

Explanation:

Step1: Calculate the probability that all 4 calls are for medical help

The probability of a single call being for medical help is \(p = 0.81\). Since the calls are independent (in the random - selection case), the probability that all \(n = 4\) calls are for medical help is given by the multiplication rule for independent events.
The formula is \(P(X = 4)=p\times p\times p\times p=p^{n}\).
Substitute \(p = 0.81\) and \(n = 4\) into the formula: \(P(X = 4)=(0.81)^{4}=0.81\times0.81\times0.81\times0.81 = 0.43046721\approx0.4305\).

Step2: Calculate the probability that at least 1 of the calls is not for medical help

The complement of the event “at least 1 call is not for medical help” is the event “all 4 calls are for medical help”.
Let \(A\) be the event “at least 1 call is not for medical help” and \(B\) be the event “all 4 calls are for medical help”. Then \(P(A)=1 - P(B)\).
We know from part (a) that \(P(B)\approx0.4305\).
So \(P(A)=1 - 0.4305 = 0.5695\).

Step3: Explain the non - validity for consecutive calls

The calculation in part (a) uses the assumption of independence. If we choose 4 consecutive calls, the 4 consecutive calls being medical may not be independent events. For example, if there is a large - scale accident, there may be a cluster of medical calls, and the outcome of one call (whether it is a medical call) may affect the outcome of the next call.

Answer:

(a) \(0.4305\)
(b) \(0.5695\)
(c) independent