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? Please help me...
To find the chance of drawing two cards with a sum of nine, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Let's break down the problem step by step:
1. Total number of possible outcomes:
- You have a stack of cards numbered 1 through 10, so there are 10 cards in total.
- For each of the two draws, you have 10 possible options since you replace the card after each draw.
- Therefore, the total number of possible outcomes is 10 * 10 = 100.
2. Number of favorable outcomes:
- To draw two cards with a sum of nine, we need to examine all the possible combinations of two cards.
- We can list the favorable combinations: (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1).
- Notice that (5,4) and (4,5) are considered separate combinations since the order matters because we replace the card after each draw.
- Therefore, there are 8 favorable outcomes.
3. Calculating chance:
- The chance (probability) of an event is the number of favorable outcomes divided by the total number of possible outcomes.
- So, the chance of drawing two cards with a sum of nine is 8/100, which simplifies to 0.08 or 8%.
In conclusion, there is an 8% chance of drawing two cards with a sum of nine from the stack of cards numbered 1 through 10 and replacing the card after each draw.