Chapter 5.1, Problem 23P

Business: Customers John runs a computer software store. Yesterday hecounted 127 people who walked by his store, 58 of whom came into the store.Of the 58, only 25 bought something in the store.

(a) Estimate the probability that a person who walks by the store will enter the store.

(b) Estimate the probability that a person who walks into the store will buy something.

(c) Estimate the probability that a person who walks by the store will come in and buy something.

(d) Estimate the probability that a person who comes into the store will buy nothing.

To determine

To find: The estimated probability that a person, who walks by the store, will enter the store.

Answer to Problem 23P

Solution: The estimated probability is 0.46.

Explanation of Solution

Given: 127 people walked by the store, 58 of them entered the store. Out of 58, 25 of them bought something.

Calculation:

The provided values can be tabulated as follows:

$\begin{array}{cccc}& \text{Buy}& \text{Did not buy}& \text{Row total}\\ \text{Came into the store}& 25& 58-25=33& 58\\ \begin{array}{l}\text{Did not come}\hfill \end{array}& 0& 69& 127-58=69\\ \text{Column total}& 25& 102& 127\end{array}$

The formula that is used to assign probability is:

$\text{Probability of an event = }\frac{\text{Number of outcomes favorable to event}}{\text{Total number of outcomes}}$

Substitute the provided values in the above formula to find the desired value of probability.

The required probability can be calculated as follows:

$\begin{array}{c}P\left(\text{Enter the store}\right)\text{= }\frac{\text{Number of person who enter the store}}{\text{Total number of person who walk by the store}}\\ =\frac{58}{127}\\ =0.4566\\ \approx 0.46\end{array}$

The required probability is 0.46.

Interpretation: There is a 46% chance that a person, who walks by the store, will enter the store.

To determine

To find: The estimated probability $P\left(\text{Buy something after entering}\right)$.

Answer to Problem 23P

Solution: The estimated probability is 0.43.

Explanation of Solution

Given: 127 people walked by the store, 58 of them came into the store. Out of 58, 25 of them bought something.

Calculation:

The provided values can be tabulated as follows:

$\begin{array}{cccc}& \text{Buy}& \text{Did not buy}& \text{Row total}\\ \text{Came into the store}& 25& 58-25=33& 58\\ \begin{array}{l}\text{Did not come}\hfill \end{array}& 0& 69& 127-58=69\\ \text{Column total}& 25& 102& 127\end{array}$

The formula that is used to assign probability is:

$\text{Probability of an event = }\frac{\text{Number of outcomes favorable to event}}{\text{Total number of outcomes}}$

Substitute the provided values in the above formula to find the desired value of probability.

The required probability can be calculated as follows:

$\begin{array}{c}P\left(\text{Buy something after entering}\right)\text{= }\frac{\text{Number of person who buy something after entering}}{\text{Total number of person who enter the store}}\\ =\frac{25}{58}\\ =0.4310\\ \approx 0.43\end{array}$

The required probability is 0.43.

Interpretation: There is a 43% chance that a person, who walks into the store, will buy something.

To determine

To find: The estimated probability $P\left(\text{Enter the store and buy something}\right)$.

Answer to Problem 23P

Solution: The estimated probability value is 0.20.

Explanation of Solution

Given: Consider the calculated probabilities from part (a) and (b). The values are: $P\left(\text{Enter the store}\right)\text{=}0.46$ and $P\left(\text{Buy something after entering}\right)\text{=0}\text{.43}$.

Calculation: Consider any two independent events A and B. The formula to calculate the probability that two independent events will occur together is given as,

$P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$

Here, entering the store and buying something after entering are independent events.

So, the required probability can be calculated as follows:

$\begin{array}{c}P\left(\text{Enter the store and buy something}\right)\text{=}P\left(\text{Enter the store}\right)\times P\left(\text{Buy something after entering}\right)\\ =0.46\times 0.43\\ =0.1978\\ \approx 0.20\end{array}$

The required probability is 0.20.

Interpretation: There is a 20% chance that a person who walks by the store will come in and buy something from the store.

To determine

To find: The estimated probability $P\left(\text{Buy nothing after entering the store}\right)$.

Answer to Problem 23P

Solution: The estimated probability is 0.57.

Explanation of Solution

Given: 127 people walked by the store, 58 of them came into the store. Out of 58, 25 of them bought something.

Calculation:

The provided values can be tabulated as follows:

$\begin{array}{cccc}& \text{Buy}& \text{Did not buy}& \text{Row total}\\ \text{Came into the store}& 25& 58-25=33& 58\\ \begin{array}{l}\text{Did not come}\hfill \end{array}& 0& 69& 127-58=69\\ \text{Column total}& 25& 102& 127\end{array}$

The formula that is used to assign probability is:

$\text{Probability of an event = }\frac{\text{Number of outcomes favorable to event}}{\text{Total number of outcomes}}$

Substitute the provided values in the above formula to find the desired value of probability.

The required probability can be calculated as follows:

$\begin{array}{c}P\left(\text{Buy nothing after entering the store}\right)\text{= }\frac{\text{Number of person who buy nothing after entering the store}}{\text{Total number of person who enter the store}}\\ \text{= }\frac{33}{58}\\ =0.5689\\ \approx 0.57\end{array}$

The required probability is 0.57.

Interpretation: There is a 57% chance that a person who comes into the store will buy nothing.