Assume that there are 16 frozen dinners: 8 pasta, 4 chicken, and 4 seafood dinners. The student selects 5 of them.
What is the probability that at least 2 of the dinners selected are pasta dinners?
I tried this:
(c(8,2)c(8,3)+c(8,3)c(8,2)+c(8,4)c(8,1))/c(16,5) = 3696/4368
But this is incorrect. I have no idea what other way to solve this. Any sugguestions?
You forgot to include c(8,5)c(8,0) in the summation. You can also compute the complementary probability of 1 or less first. Then you only need to sum over two terms.
To find the probability that at least 2 of the dinners selected are pasta dinners, we need to consider the different combinations that satisfy this condition.
First, let's calculate the total number of ways the student can select any 5 dinners from the 16 available. This is given by the combination formula: C(16, 5) = 4368, as you correctly calculated.
Now, let's break down the different cases that satisfy the condition of at least 2 pasta dinners:
Case 1: Selecting exactly 2 pasta dinners and 3 non-pasta dinners.
The number of ways to choose 2 pasta dinners from 8 available is C(8, 2) = 28.
And the number of ways to choose 3 non-pasta dinners from the remaining 8 available is C(8, 3) = 56.
So, the total number of combinations for this case is 28 * 56 = 1568.
Case 2: Selecting exactly 3 pasta dinners and 2 non-pasta dinners.
The number of ways to choose 3 pasta dinners from 8 available is C(8, 3) = 56.
And the number of ways to choose 2 non-pasta dinners from the remaining 8 available is C(8, 2) = 28.
So, the total number of combinations for this case is 56 * 28 = 1568.
Case 3: Selecting exactly 4 pasta dinners and 1 non-pasta dinner.
The number of ways to choose 4 pasta dinners from 8 available is C(8, 4) = 70.
And the number of ways to choose 1 non-pasta dinner from the remaining 8 available is C(8, 1) = 8.
So, the total number of combinations for this case is 70 * 8 = 560.
To find the total number of combinations that satisfy the condition, we need to sum up the number of combinations from each case:
Total number of combinations = 1568 + 1568 + 560 = 3696.
Finally, divide the total number of combinations by the total number of ways to select 5 dinners (4368) to get the probability:
Probability = 3696 / 4368 = 12 / 14 = 6 / 7 ≈ 0.8571.
Therefore, the probability that at least 2 of the dinners selected are pasta dinners is approximately 0.8571, or 85.71%.