Question

1. customer purchases in a department store there are 120 customers, 90 of whom will buy at least 1 item. if 5 customers are selected at random, one by one, find the probability that all will buy at least 1 item.

Explanation:

Step1: Calculate the number of non - buyers

The number of customers who will not buy at least 1 item is \(120 - 90=30\).

Step2: Calculate the probability for the first selection

The probability that the first customer selected will buy at least 1 item is \(\frac{90}{120}\).

Step3: Calculate the probability for the second selection

After one buyer is selected, there are 89 buyers left out of 119 customers. So the probability that the second customer selected will buy at least 1 item is \(\frac{89}{119}\).

Step4: Calculate the probability for the third selection

After two buyers are selected, there are 88 buyers left out of 118 customers. So the probability that the third customer selected will buy at least 1 item is \(\frac{88}{118}\).

Step5: Calculate the probability for the fourth selection

After three buyers are selected, there are 87 buyers left out of 117 customers. So the probability that the fourth customer selected will buy at least 1 item is \(\frac{87}{117}\).

Step6: Calculate the probability for the fifth selection

After four buyers are selected, there are 86 buyers left out of 116 customers. So the probability that the fifth customer selected will buy at least 1 item is \(\frac{86}{116}\).

Step7: Calculate the overall probability

The probability that all 5 customers will buy at least 1 item is the product of the probabilities of each individual selection:

$$P=\frac{90}{120}\times\frac{89}{119}\times\frac{88}{118}\times\frac{87}{117}\times\frac{86}{116}$$

$$P=\frac{90\times89\times88\times87\times86}{120\times119\times118\times117\times116}$$

$$P=\frac{553541760}{19073496960}$$

$$P=\frac{553541760\div 48}{19073496960\div 48}=\frac{11532120}{397364520}$$

$$P=\frac{11532120\div 12}{397364520\div 12}=\frac{961010}{33113710}$$

$$P=\frac{961010\div 10}{33113710\div 10}=\frac{96101}{3311371}\approx0.029$$

Answer:

The probability that all 5 customers will buy at least 1 item is approximately \(0.029\)