Drawing 5 cards out of 13 cards numbered 1 to 13 without replacement, what is the probability of having a sum greater than 40 ?
To find the probability of drawing 5 cards out of 13 cards numbered 1 to 13 without replacement and getting a sum greater than 40, we need to break down the problem into smaller steps.
Step 1: Determine the total number of possible outcomes.
Since we are drawing 5 cards without replacement, the total number of possible outcomes can be calculated using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n represents the total number of cards (13) and r represents the number of cards drawn (5):
C(13, 5) = 13! / (5! * (13 - 5)!)
Step 2: Determine the number of favorable outcomes.
To find the number of favorable outcomes, we need to count the number of ways we can draw 5 cards with a sum greater than 40. We can do this by considering the possible values for the drawn cards and their respective combinations.
Cards 9, 10, 11, 12, 13:
There is only one combination of these cards that has a sum greater than 40, which is {9, 10, 11, 12, 13}.
So, the number of favorable outcomes for this combination is 1.
Cards 8, 9, 10, 11, 12:
Again, there is only one combination of these cards that has a sum greater than 40, which is {8, 9, 10, 11, 12}.
So, the number of favorable outcomes for this combination is also 1.
Cards 7, 8, 9, 10, 11:
The only combination of these cards that has a sum greater than 40 is {7, 8, 9, 10, 11}.
So, the number of favorable outcomes for this combination is 1.
Cards 6, 7, 8, 9, 10:
The only combination of these cards that has a sum greater than 40 is {6, 7, 8, 9, 10}.
So, the number of favorable outcomes for this combination is 1.
Cards 5, 6, 7, 8, 9:
The only combination of these cards that has a sum greater than 40 is {5, 6, 7, 8, 9}.
So, the number of favorable outcomes for this combination is 1.
Therefore, the total number of favorable outcomes is 1 + 1 + 1 + 1 + 1 = 5.
Step 3: Calculate the probability.
To calculate the probability, we divide the number of favorable outcomes (5) by the total number of possible outcomes (C(13, 5)):
Probability = favorable outcomes / total outcomes
Probability = 5 / C(13, 5)
Simplifying C(13, 5):
C(13, 5) = 13! / (5! * (13 - 5)!)
C(13, 5) = 13! / (5! * 8!)
C(13, 5) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)
C(13, 5) = 1287
Therefore, the probability of drawing 5 cards without replacement and getting a sum greater than 40 is:
Probability = 5 / 1287