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? I have to use the charts for the possibilities:
First Card
1 2 3 4 5 6 7 8 9 10
s 1
e 2
c 3
o 4
n 5
d 6
c 7
a 8
r 9
d 10
there are 8 ways to get a sum of 9
1 8, 2 7, .... , 8,1
let's look at the prob of getting one of those pairs, the 1 8
prob of getting the 1 is 1/10.
since you are replacing the card, the prob of getting an 8 on the second draw is also 1/10
so the prob of getting the 1 8 combination is (1/10)(1/10) = 1/100
but there are 8 of those cases
so the prob of getting a sum of 9 when drawing 2 cards with replacement is 8/100 = 2/25
I want to play this game!
2/25
Ah, I see you've got your handy-dandy chart there! Let's dive into the math and see what we can come up with.
So, to find the chance of drawing two cards with a sum of nine, we need to look at all the possible combinations. From the chart, we can see that the only combination that adds up to nine is (3,6) or (6,3).
Now, since you replace the first card and shuffle the deck again, we essentially have two independent events. That means the probability of drawing a 3 on the first draw is 1/10, and the probability of drawing a 6 on the second draw is also 1/10.
To calculate the chance of both events happening, you multiply the individual probabilities together. So, the probability of drawing two cards with a sum of nine is (1/10) * (1/10) = 1/100.
So, my friend, the chance of you winning that $1 million is 1 in 100! Good luck, and may the cards be ever in your favor!
To calculate the chance of drawing two cards with a sum of nine, we need to find all the possible combinations of the first and second card that add up to nine. Let's go step by step.
1. Look at the chart and find the row labeled "First Card" and the column labeled "Second Card."
2. Start by finding the number 1 in the "First Card" row. Then, look for the numbers in the "Second Card" column that add up to a sum of nine with 1. In this case, there is only one possibility: 8. So, put a checkmark or mark it with an "x" in the corresponding cell where the row and column intersect (1,8).
3. Repeat this process for the rest of the numbers in the "First Card" row. Here are all the possible combinations:
- (1,8)
- (2,7)
- (3,6)
- (4,5)
- (5,4)
- (6,3)
- (7,2)
- (8,1)
4. Count the total number of possible combinations that add up to nine. In this case, there are eight possible combinations.
5. Since you replace the card and shuffle the stack again for the second draw, the chances of drawing any number for the second card remain the same as for the first card.
6. Calculate the overall probability by dividing the number of successful outcomes (combinations that add up to nine) by the total number of possible outcomes (all the combinations possible when drawing two cards). In this case, there are ten possible cards to choose from for each draw, so the total number of possible outcomes is 10 * 10 = 100.
7. Therefore, the chance of drawing two cards with a sum of nine is 8/100 or 8%.
To determine the chance of drawing two cards with a sum of nine, we can use the chart you provided. Let me explain step-by-step how to find the answer using the chart:
1. Determine the number of ways to select the first card that would result in a sum of nine.
- Looking at the chart, we can see that there are two possibilities: drawing a card with a value of 2 or drawing a card with a value of 3.
2. Determine the number of ways to select the second card that would result in a sum of nine.
- Since you replace the first card and shuffle the stack again, each card has an equal chance to be drawn. Therefore, there are 10 possibilities for the second card.
3. Calculate the total number of possible outcomes for the two-card draws.
- In this case, you have 10 options for the first card and 10 options for the second card, resulting in a total of 10 x 10 = 100 possible outcomes.
4. Calculate the number of favorable outcomes.
- Since there are two ways to select the first card and 10 ways to select the second card, there are 2 x 10 = 20 favorable outcomes.
5. Determine the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
- The probability is 20 favorable outcomes out of a total of 100 possible outcomes, which simplifies to 20/100 or 1/5.
Therefore, the chance that your two cards will have a sum of nine is 1/5 or 20%.