Question

A new restaurant holds a free tasting event to generate business. It sends invitations to members of a local gourmet eating club. It also has staff stand outside during the event to encourage people walking by who may not have received an invitation to attend. As guests are leaving the free event, some make dinner reservations for a future date and some do not. The table below shows the results.

Results of Free Tasting Event
Received an Invitation Did Not Receive an Invitation
Made a Reservation 27 21
Did Not Make a Reservation 48 84
Based on this data, what is the probability that a randomly selected guest made a reservation given that the guest did not receive an invitation?

A.
0.20


B.
0.27


C.
0.36


D.
0.44

Answer 1

To find the probability that a randomly selected guest made a reservation given that the guest did not receive an invitation, we can use the formula for conditional probability:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

where:
- \( A \) is the event that a guest made a reservation,
- \( B \) is the event that a guest did not receive an invitation.

From the table, we have the following data for guests who did not receive an invitation:

- Made a reservation: 21
- Did not make a reservation: 84

First, we need to determine the total number of guests who did not receive an invitation:

\[
\text{Total who did not receive an invitation} = 21 + 84 = 105
\]

Now, we can determine the conditional probability \( P(A | B) \):

\[
P(A | B) = \frac{\text{Number of guests who made a reservation and did not receive an invitation}}{\text{Total number of guests who did not receive an invitation}} = \frac{21}{105}
\]

Now we calculate \( \frac{21}{105} \):

\[
\frac{21}{105} = \frac{1}{5} = 0.20
\]

Thus, the probability that a randomly selected guest made a reservation given that the guest did not receive an invitation is:

**A. 0.20**