Question

according to sales records at a local coffee shop, 75% of all customers like hot coffee, 30% like iced coffee, and 22% like both hot and iced coffee. the venn diagram displays the coffee preferences of the customers. a randomly selected customer is asked if they like hot or iced coffee. let h be the event that the customer likes hot coffee and let / be the event that the customer likes iced coffee. what is the probability that a randomly selected customer likes hot or iced coffee? 0.22 0.30 0.61 0.83

Explanation:

Step1: Recall the formula for \(P(H\cup I)\)

The formula for the probability of the union of two events \(A\) and \(B\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Here, \(A = H\) (customer likes hot coffee), \(P(H) = 0.75\); \(B = I\) (customer likes iced coffee), \(P(I)=0.30\); and \(P(H\cap I) = 0.22\).

Step2: Substitute the values into the formula

Substitute the values into the formula \(P(H\cup I)=0.75 + 0.30-0.22\).
First, add \(0.75\) and \(0.30\): \(0.75+0.30=\frac{75 + 30}{100}=\frac{105}{100}=1.05\).
Then subtract \(0.22\) from \(1.05\): \(1.05-0.22=\frac{105 - 22}{100}=\frac{83}{100}=0.83\).
Another way is to use the Venn - diagram values. The probability \(P(H\cup I)\) is the sum of the non - overlapping parts of \(H\) and \(I\) and the overlapping part. That is \(0.53+0.22 + 0.08=(0.53+0.08)+0.22=0.61+0.22 = 0.83\).

Answer:

\(0.83\)