You order sixteen burritos to go from a Mexican restaurant: seven with hot peppers and nine without. However, the restaurant forgot to label them. If you pick four burritos at random, find the probability that at least two of the burritos you pick have hot peppers.
To solve this problem, we need to consider the possible combinations of burritos that satisfy the condition of at least two having hot peppers.
First, let's determine the total number of ways to pick 4 burritos out of 16 without any restrictions. This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
In our case, n = 16 (total number of burritos) and r = 4 (number of burritos we want to pick). Plugging in the values:
C(16, 4) = 16! / (4!(16-4)!)
= 16! / (4!12!)
Next, we need to calculate the number of combinations where none of the picked burritos have hot peppers. Since there are 9 burritos without hot peppers, we need to select all 4 burritos from this group. Using the combination formula again:
C(9, 4) = 9! / (4!(9-4)!)
Now, we can find the number of combinations where only one of the picked burritos has hot peppers. We need to choose 3 burritos from the 7 with hot peppers and 1 burrito from the remaining 9 without hot peppers. Again, using the combination formula:
C(7, 3) * C(9, 1) = (7! / (3!(7-3)!)) * (9! / (1!(9-1)!))
Finally, we can calculate the probability of picking at least two burritos with hot peppers by taking the complement of the probability of having zero or one burrito with hot peppers:
P(at least 2 burritos with hot peppers) = 1 - P(0 burritos with hot peppers) - P(1 burrito with hot peppers)
To find the probability, divide each of the values calculated above by the total number of combinations C(16, 4).