Assume that the playbook contains 11 passing plays and 15 running plays. The coach randomly selects 8 plays from the playbook.
What is the probability that the coach selects at least 2 passing plays and at least 3 running plays?
Add up these probabilities:
2 running (5 passing)
1 passing (7 running)
1 running (7 passing)
0 running (8 passing)
0 passing (8 running)
and subtract the total from 1.
The probability of 0 passing plays is (15/26)^8 = 0.01227
You do the others
To find the probability, we need to calculate the total number of possible combinations and the number of desired combinations.
First, let's calculate the total number of possible combinations. Since the coach randomly selects 8 plays from the playbook, we can use the formula for combinations:
nCr = n! / (r! * (n-r)!)
In this case, n represents the total number of plays in the playbook, which is 11 + 15 = 26, and r represents the number of plays the coach selects, which is 8.
So, the total number of possible combinations is:
26C8 = 26! / (8! * (26-8)!)
Next, let's calculate the number of desired combinations. We want the coach to select at least 2 passing plays and at least 3 running plays.
Case 1: Selecting exactly 2 passing plays and 3 running plays
The total number of ways to choose 2 passing plays from the 11 available is 11C2, and the total number of ways to choose 3 running plays from the 15 available is 15C3.
Case 2: Selecting exactly 2 passing plays and 4 running plays
The total number of ways to choose 2 passing plays from the 11 available is 11C2, and the total number of ways to choose 4 running plays from the 15 available is 15C4.
Case 3: Selecting exactly 2 passing plays and 5 running plays
The total number of ways to choose 2 passing plays from the 11 available is 11C2, and the total number of ways to choose 5 running plays from the 15 available is 15C5.
Case 4: Selecting exactly 3 passing plays and 3 running plays
The total number of ways to choose 3 passing plays from the 11 available is 11C3, and the total number of ways to choose 3 running plays from the 15 available is 15C3.
Case 5: Selecting exactly 3 passing plays and 4 running plays
The total number of ways to choose 3 passing plays from the 11 available is 11C3, and the total number of ways to choose 4 running plays from the 15 available is 15C4.
Now, let's calculate the number of desired combinations by adding up all these different cases.
Number of desired combinations = (11C2 * 15C3) + (11C2 * 15C4) + (11C2 * 15C5) + (11C3 * 15C3) + (11C3 * 15C4)
Finally, the probability of selecting at least 2 passing plays and at least 3 running plays is:
Probability = (Number of desired combinations) / (Total number of possible combinations)
I will now calculate these values for you.
To find the probability of selecting at least 2 passing plays and at least 3 running plays, we need to calculate the probability of selecting 2 or more passing plays and 3 or more running plays, and then subtract the probability of selecting fewer than 2 passing plays or fewer than 3 running plays.
Let's break it down step by step:
1. Calculate the probability of selecting 2 or more passing plays:
- There are 11 passing play options in the playbook.
- Since the coach selects 8 plays, we need to consider combinations of choosing 2, 3, 4, 5, 6, 7, and 8 passing plays.
- To calculate the probability, we divide the number of combinations by the total number of possible selections, which is the sum of combinations of choosing any 8 plays from the entire playbook (11 passing plays + 15 running plays).
- We can use the formula for combinations: C(n, r) = (n!)/(r!(n-r)!), where n is the total number of options and r is the number of choices.
- So the probability of selecting 2 or more passing plays is the sum of the probabilities for each combination.
2. Calculate the probability of selecting 3 or more running plays:
- Similarly, there are 15 running play options in the playbook.
- We need to consider combinations of choosing 3, 4, 5, 6, 7, and 8 running plays.
- Again, we divide the number of combinations by the total number of possible selections.
3. Calculate the probability of selecting fewer than 2 passing plays or fewer than 3 running plays:
- To calculate this, we need to subtract the probabilities calculated in steps 1 and 2 from 1, as it covers the opposite case.
4. Calculate the final probability:
- Finally, we subtract the probability of not selecting at least 2 passing plays or not selecting at least 3 running plays from 1 to get the desired probability.
Let's calculate these probabilities step by step.