To win $1 million, you must draw two cards whose sum is nine from a stack of cards numbered 1 through 10. After the first draw, you replace the card and shuffle the stack again for the second draw. What is the chance that your two cards will have a sum of nine?
The chance that your two cards will have a sum of nine is 1/45, or 2.22%.
To calculate the chance of drawing two cards with a sum of nine, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Favorable outcomes: To have a sum of nine, the possible combinations of two cards are as follows:
- Cards (3, 6): One card with a value of 3 and another card with a value of 6.
- Cards (4, 5): One card with a value of 4 and another card with a value of 5.
- Cards (5, 4): One card with a value of 5 and another card with a value of 4.
- Cards (6, 3): One card with a value of 6 and another card with a value of 3.
Therefore, there are four favorable outcomes.
Total possible outcomes: Since there are 10 cards numbered 1 to 10, the total number of possible outcomes for the first draw is 10. After the first draw, the deck is shuffled again for the second draw, resulting in another 10 possible outcomes.
Since the two draws are independent events, the total number of possible outcomes for both draws is the product of the number of outcomes for each draw, which is 10 * 10 = 100.
Therefore, the chance of drawing two cards with a sum of nine is: favorable outcomes / total possible outcomes = 4 / 100 = 0.04 or 4%.
To determine the chance of drawing two cards with a sum of nine, we need to calculate the probability for each individual draw and then multiply them.
Let's break it down:
1. Probability of drawing a card with a value that adds up to nine on the first draw:
- To get a sum of nine, you can draw these pairs: (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1).
- There are 8 possible pairs out of a total of 100 combinations (10 cards * 10 cards).
- Therefore, the probability of drawing a card with a value that adds up to nine on the first draw is 8/100 or 0.08 (8%).
2. Probability of drawing a card with a value that adds up to nine on the second draw:
- Since you replace the card and shuffle the stack, the probability remains the same as the first draw.
- Therefore, the probability of drawing a card with a value that adds up to nine on the second draw is also 8/100 or 0.08 (8%).
3. Probability of both events happening:
- To determine the probability of two independent events happening, we multiply their individual probabilities together.
- So, multiplying the probabilities from the first and second draws, we get (8/100) * (8/100) = 64/10,000 or 0.0064 (0.64%).
Therefore, the chance that your two cards will have a sum of nine is 0.0064 which is equivalent to 0.64%.