Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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.
### 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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
