Sure, let's solve this step-by-step.
### Problem Setup:
Bill and Ben each have three cards numbered 1, 2, and 3. They each pick one card and then multiply the numbers on their chosen cards.
### Part (a): Probability that the product is an even number
Step-by-Step Solution:
1. List all possible outcomes:
Let's list all the combinations of the cards they can pick:
- Bill picks 1, Ben picks 1: [tex]\(1 \times 1 = 1\)[/tex]
- Bill picks 1, Ben picks 2: [tex]\(1 \times 2 = 2\)[/tex]
- Bill picks 1, Ben picks 3: [tex]\(1 \times 3 = 3\)[/tex]
- Bill picks 2, Ben picks 1: [tex]\(2 \times 1 = 2\)[/tex]
- Bill picks 2, Ben picks 2: [tex]\(2 \times 2 = 4\)[/tex]
- Bill picks 2, Ben picks 3: [tex]\(2 \times 3 = 6\)[/tex]
- Bill picks 3, Ben picks 1: [tex]\(3 \times 1 = 3\)[/tex]
- Bill picks 3, Ben picks 2: [tex]\(3 \times 2 = 6\)[/tex]
- Bill picks 3, Ben picks 3: [tex]\(3 \times 3 = 9\)[/tex]
2. Count the number of possible outcomes:
There are [tex]\(3 \times 3 = 9\)[/tex] possible outcomes since each player has three choices independently.
3. Identify the even outcomes:
- [tex]\(1 \times 2 = 2\)[/tex]
- [tex]\(2 \times 1 = 2\)[/tex]
- [tex]\(2 \times 2 = 4\)[/tex]
- [tex]\(2 \times 3 = 6\)[/tex]
- [tex]\(3 \times 2 = 6\)[/tex]
These are 5 outcomes where the product is even.
4. Calculate the probability:
[tex]\[
P(\text{even product}) = \frac{\text{number of even outcomes}}{\text{total number of outcomes}} = \frac{5}{9}
\][/tex]
So, the probability that the product is an even number is [tex]\(\frac{5}{9}\)[/tex].
### Part (b): Probability that the product is greater than 5
Step-by-Step Solution:
1. List all possible outcomes (reused from Part (a)):
- Bill picks 1, Ben picks 1: [tex]\(1 \times 1 = 1\)[/tex]
- Bill picks 1, Ben picks 2: [tex]\(1 \times 2 = 2\)[/tex]
- Bill picks 1, Ben picks 3: [tex]\(1 \times 3 = 3\)[/tex]
- Bill picks 2, Ben picks 1: [tex]\(2 \times 1 = 2\)[/tex]
- Bill picks 2, Ben picks 2: [tex]\(2 \times 2 = 4\)[/tex]
- Bill picks 2, Ben picks 3: [tex]\(2 \times 3 = 6\)[/tex]
- Bill picks 3, Ben picks 1: [tex]\(3 \times 1 = 3\)[/tex]
- Bill picks 3, Ben picks 2: [tex]\(3 \times 2 = 6\)[/tex]
- Bill picks 3, Ben picks 3: [tex]\(3 \times 3 = 9\)[/tex]
2. Identify the outcomes greater than 5:
- [tex]\(2 \times 3 = 6\)[/tex]
- [tex]\(3 \times 2 = 6\)[/tex]
- [tex]\(3 \times 3 = 9\)[/tex]
These are 3 outcomes where the product is greater than 5.
3. Calculate the probability:
[tex]\[
P(\text{product > 5}) = \frac{\text{number of outcomes > 5}}{\text{total number of outcomes}} = \frac{3}{9} = \frac{1}{3}
\][/tex]
So, the probability that the product is greater than 5 is [tex]\(\frac{1}{3}\)[/tex].
