Question

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.

b. What’s the probability that at least $1$of the calls is not for medical help?

c. Explain why the calculation in part (a) may not be valid if we choose 4 consecutive calls to the station.

Short answer

Part a. Probability for the randomly chosen all 4 calls are for medical help is approx. $0.43305$

Part b. Probability that at least $1$of the calls is not for medical help is $0.5695$.

Part c. It is not necessary that calls are independent of each other.

Step-by-step solution

Part a. Step 1. Given information

$81\%$of the incoming calls are for medical help.

$4$incoming calls to the station are chosen at random.

Part a. Step 2. Calculation

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to multiplication rule for independent events,

$P(AandB)=P(A\cap B)=P\left(A\right)\times P\left(B\right)$

Let

A: One incoming call is for medical help

B: $4$incoming calls are for medical help

Now,

Probability for the incoming call is for medical help,

$P\left(A\right)=81\%=0.81$

Since the incoming calls are selected at random, it would be more convenient to assume that incoming calls are independent of each other.

Thus,

For probability that $4$incoming calls are for medical help, apply multiplication rule for independent events:

role="math" localid="1664275493508" $$\begin{gathered} P\left(B\right)=P\left(A\right)\times P\left(A\right)\times P\left(A\right)\times P\left(A\right) \\ =(P{\left(A\right))}^4 \\ ={(0.81)}^4 \\ \approx 0.4305 \end{gathered}$$

Thus,

Probability for the randomly selected all $4$incoming calls are for medical help is approx.$0.4305$

Part b. Step 1. Calculation

According to complement rule,

$P\left(A^c\right)=P(notA)=1-P\left(A\right)$

Let

$B$4 incoming calls are for medical help

$B^c:$None of the $4$incoming calls are for medical help

From Part (a),

We have

Probability for randomly selected all 4 incoming calls are for medical help,

$P\left(B\right)\approx 0.4305$

We have of find the probability for at least $1$of the calls is not for medical help.

That means

None of the $4$incoming calls is for medical help.

Apply the complement rule:

$$\begin{gathered} P\left(B^c\right)=1-P\left(B\right) \\ =1-0.4305 \\ =0.5695 \end{gathered}$$

Thus,

Probability that at least $1$of the calls is not for medical help is$0.5695$

Part c. Step 1. Calculation

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to multiplication rule for independent events,

$P(AandB)=P(A\cap B)=P\left(A\right)\times P\left(B\right)$

In Part (a),

Multiplication rule for independent events has been used.

When $4$consecutive calls are chosen, there may be some possibility that occurrence of these $4$calls would be same.

That means

$4$consecutive calls may be for the same accident when compared to the randomly chosen $4$calls.

Thus,

This will affect the probability for medical call if these $4$calls are about the same occurrence.

This implies

The incoming calls will be no longer independent.

Thus,

Use of the multiplication for independent events would be inappropriate.