Question

on a particular day, a restaurant that is open for lunch and dinner had 119 customers. each customer came in for one meal. an employee recorded at which meal each customer came in and whether the customer ordered dessert. the data are summarized in the table below.

dessert no dessert
lunch 21 9
dinner 37 52

suppose a customer from that day is chosen at random.
answer each part. do not round intermediate computations, and round your answers to the nearest hundredth.

(a) what is the probability that the customer did not order dessert?

(b) what is the probability that the customer came for lunch or did not order dessert?

Explanation:

Part (a)

Step1: Find total no - dessert customers

To find the number of customers who did not order dessert, we add the number of lunch customers who did not order dessert and dinner customers who did not order dessert. From the table, lunch no - dessert is 9 and dinner no - dessert is 52. So, the number of customers who did not order dessert is \(9 + 52=61\).

Step2: Calculate the probability

The total number of customers is 119. The probability \(P(\text{no dessert})\) is the number of customers who did not order dessert divided by the total number of customers. So, \(P(\text{no dessert})=\frac{61}{119}\approx0.51\) (rounded to the nearest hundredth).

Step1: Recall the addition rule of probability

The addition rule for two events \(A\) and \(B\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event that the customer came for lunch and \(B\) be the event that the customer did not order dessert.

Step2: Find \(P(A)\)

The number of lunch customers is \(21 + 9 = 30\). So, \(P(A)=\frac{30}{119}\).

Step3: Find \(P(B)\)

We already found in part (a) that \(P(B)=\frac{61}{119}\).

Step4: Find \(P(A\cap B)\)

The number of customers who came for lunch and did not order dessert is 9. So, \(P(A\cap B)=\frac{9}{119}\).

Step5: Calculate \(P(A\cup B)\)

Using the addition rule:
\[

$$\begin{align*} P(A\cup B)&=\frac{30}{119}+\frac{61}{119}-\frac{9}{119}\\ &=\frac{30 + 61-9}{119}\\ &=\frac{82}{119}\approx0.69 \end{align*}$$

\]

Answer:

\(0.51\)

Part (b)