To find the probability that the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is more than 20, where [tex]\( x \)[/tex] is chosen from the set \{1, 2, 3, 4\} and [tex]\( y \)[/tex] is chosen from the set \{5, 6, 7, 8\}, we can follow these steps:
### Step 1: Determine the Total Number of Possible Outcomes
Each choice of [tex]\( x \)[/tex] can be paired with each choice of [tex]\( y \)[/tex]. Since there are 4 choices for [tex]\( x \)[/tex] and 4 choices for [tex]\( y \)[/tex], the total number of possible outcomes is:
[tex]\[ 4 \times 4 = 16 \][/tex]
### Step 2: Identify the Favorable Outcomes
We need to count the number of combinations where the product [tex]\( xy \)[/tex] is greater than 20. We will evaluate this condition for each possible pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- For [tex]\( x = 1 \)[/tex]:
- [tex]\( 1 \times 5 = 5 \)[/tex] (not greater than 20)
- [tex]\( 1 \times 6 = 6 \)[/tex] (not greater than 20)
- [tex]\( 1 \times 7 = 7 \)[/tex] (not greater than 20)
- [tex]\( 1 \times 8 = 8 \)[/tex] (not greater than 20)
- For [tex]\( x = 2 \)[/tex]:
- [tex]\( 2 \times 5 = 10 \)[/tex] (not greater than 20)
- [tex]\( 2 \times 6 = 12 \)[/tex] (not greater than 20)
- [tex]\( 2 \times 7 = 14 \)[/tex] (not greater than 20)
- [tex]\( 2 \times 8 = 16 \)[/tex] (not greater than 20)
- For [tex]\( x = 3 \)[/tex]:
- [tex]\( 3 \times 5 = 15 \)[/tex] (not greater than 20)
- [tex]\( 3 \times 6 = 18 \)[/tex] (not greater than 20)
- [tex]\( 3 \times 7 = 21 \)[/tex] (greater than 20)
- [tex]\( 3 \times 8 = 24 \)[/tex] (greater than 20)
- For [tex]\( x = 4 \)[/tex]:
- [tex]\( 4 \times 5 = 20 \)[/tex] (not greater than 20)
- [tex]\( 4 \times 6 = 24 \)[/tex] (greater than 20)
- [tex]\( 4 \times 7 = 28 \)[/tex] (greater than 20)
- [tex]\( 4 \times 8 = 32 \)[/tex] (greater than 20)
From these calculations, we find the favorable pairs (x, y) are:
- (3, 7)
- (3, 8)
- (4, 6)
- (4, 7)
- (4, 8)
Thus, there are 5 favorable outcomes.
### Step 3: Calculate the Probability
The probability [tex]\( P \)[/tex] of the product [tex]\( xy \)[/tex] being greater than 20 is the ratio of the number of favorable outcomes to the total number of outcomes:
[tex]\[ P(\text{xy} > 20) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{5}{16} \][/tex]
### Step 4: Convert to a Decimal (Optional for Clarity)
[tex]\[ \frac{5}{16} = 0.3125 \][/tex]
Therefore, the probability that the product [tex]\( xy \)[/tex] is more than 20 is [tex]\( \frac{5}{16} \)[/tex] or 0.3125.
