Assume that there are 13 frozen dinners: 5 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?
This is the same as probability of 2, 3, 4 or 5 dinners being selected.
P(2) = 5/13 * 4/12 * 8/11 * 7/10 * 6/9 = ?
P(5) = 5/13 * 4/12 * 3/11 * 2/10 * 1/9 = ?
Probability of P(2), P(3), P(4) or P(5) = sum of these probabilities.
To find the probability that at least 2 of the dinners selected are pasta dinners, we need to first calculate the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
The student is selecting 5 dinners out of the total of 13, so the total number of possible outcomes can be calculated using combinations. We can use the formula for combinations, denoted as "nCr", where n is the total number of items and r is the number of items to be selected. In this case, n = 13 and r = 5.
Total number of possible outcomes = 13C5
Favorable outcomes:
Now, let's calculate the number of favorable outcomes - the number of ways to select at least 2 pasta dinners.
Case 1: Selecting exactly 2 pasta dinners:
To calculate the favorable outcomes for selecting exactly 2 pasta dinners, we can use combinations again. The number of ways to select 2 pasta dinners from the available 5 is 5C2. The remaining 3 dinners can be any combination of chicken and seafood, so we also need to calculate the selection of the remaining 3 dinners from the non-pasta dinners, which is 8C3.
Favorable outcomes for exactly 2 pasta dinners = 5C2 * 8C3
Case 2: Selecting exactly 3 pasta dinners:
Similar to the previous case, the number of ways to select 3 pasta dinners is 5C3, and the remaining 2 dinners can be any combination of chicken and seafood, which is 8C2.
Favorable outcomes for exactly 3 pasta dinners = 5C3 * 8C2
Case 3: Selecting 4 pasta dinners:
The number of ways to select all 4 pasta dinners is simply 5C4. The remaining dinner can be any of the non-pasta dinners, which is 8C1.
Favorable outcomes for selecting 4 pasta dinners = 5C4 * 8C1
Case 4: Selecting all 5 pasta dinners:
Since there are only 5 pasta dinners available, the number of favorable outcomes for this case is simply 1.
Favorable outcomes for selecting all 5 pasta dinners = 1
Now, let's calculate the total number of favorable outcomes by summing up the cases:
Total number of favorable outcomes = (Favorable outcomes for exactly 2 pasta dinners) + (Favorable outcomes for exactly 3 pasta dinners) + (Favorable outcomes for selecting 4 pasta dinners) + (Favorable outcomes for selecting all 5 pasta dinners)
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Total number of favorable outcomes / Total number of possible outcomes
Substituting the values and performing the calculations will provide the final probability.