Question

Alex chooses a number at random from the set $\{1, 2, 3, \dots, 5\}.$  Winnie also chooses a number at random from the same set.  (They can choose the same number.)  What is the probability that the product of their numbers is even?

Answer 1

[AI-generated answer — Bot GPT-4o]

To determine the probability that the product of two randomly chosen numbers from the set \( \{1, 2, 3, 4, 5\} \) is even, we first identify the conditions under which the product is even. Specifically, a product is even if at least one of the numbers is even.

The set \( \{1, 2, 3, 4, 5\} \) includes these numbers:
- Odds: 1, 3, 5
- Evens: 2, 4

### Step 1: Calculate the total number of possible outcomes
Each person can choose any of 5 numbers, so the total number of pairs \((a,b)\) they can form is:
\[
5 \times 5 = 25
\]

### Step 2: Calculate the number of odd-odd pairs
The number of odd choices is 3 (1, 3, 5). Both people choosing odd numbers will result in an odd product (which is not even). The number of outcomes where both Alex and Winnie choose odd numbers is:
\[
3 \times 3 = 9
\]

### Step 3: Calculate the number of even-number pairs
An even product occurs if at least one of the numbers chosen is even. Therefore, the number of favorable pairs is the total number of pairs minus the number of odd-odd pairs. Specifically,
\[
25 - 9 = 16
\]

### Step 4: Calculate the probability
The probability \( P \) that the product is even is the ratio of favorable outcomes to the total number of outcomes:
\[
P = \frac{16}{25}
\]

Thus, the probability that the product of the two chosen numbers is even is:
\[
\boxed{\frac{16}{25}}
\]