Question

examples 2 and 3

1. awards the student of the month gets to choose one award from 9 gift certificates to area restaurants, 8 t - shirts, 6 water bottles, or 5 gift cards to the mall. what is the probability that the student of the month chooses a t - shirt or a water bottle?

Explanation:

Step1: Calculate total number of awards

To find the total number of awards, we sum up the number of each type of award. The number of gift certificates is 9, T - shirts is 8, water bottles is 6, and mall gift cards is 5. So the total number of awards \( n(S)=9 + 8+6 + 5\).
\(n(S)=9 + 8+6 + 5=28\)

Step2: Calculate number of favorable outcomes

The favorable outcomes are choosing a T - shirt or a water bottle. The number of T - shirts is 8 and the number of water bottles is 6. So the number of favorable outcomes \(n(A\cup B)=n(A)+n(B)\) (where \(A\) is the event of choosing a T - shirt and \(B\) is the event of choosing a water bottle, and since a T - shirt and a water bottle are mutually exclusive, we can add their counts).
\(n(A\cup B)=8 + 6=14\)

Step3: Calculate the probability

The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So the probability of choosing a T - shirt or a water bottle is \(P=\frac{n(A\cup B)}{n(S)}\).
Substituting the values we found: \(P = \frac{14}{28}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)