What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly check Id of five students from among nine students of which four are not of legal age ?
The correct answer and explanation is :
To solve this probability problem, let’s break it down step by step:
Given Information:
- There are 9 students in total.
- Out of these, 4 are minors (underage).
- The remaining 5 students are of legal age.
- The witness is randomly selecting 5 students to check their IDs.
- We are interested in the probability that exactly 2 minors are selected to check their IDs.
Step 1: Total number of possible selections
First, we need to calculate the total number of ways to select 5 students from a group of 9 students. This is a combination problem where we are choosing 5 students out of 9. The number of ways to do this is given by the combination formula:
[
C(n, k) = \frac{n!}{k!(n-k)!}
]
Where (n) is the total number of students, and (k) is the number of students to be selected. In this case, (n = 9) and (k = 5):
[
C(9, 5) = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
]
So, there are 126 total ways to select 5 students from 9.
Step 2: Number of favorable selections (exactly 2 minors)
Now, we need to calculate how many ways the witness can select exactly 2 minors and 3 legal students. The number of ways to select 2 minors out of the 4 minors is:
[
C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
]
Similarly, the number of ways to select 3 legal students out of the 5 legal students is:
[
C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
]
So, the total number of favorable outcomes (selecting 2 minors and 3 legal students) is the product of the two combinations:
[
6 \times 10 = 60
]
Step 3: Probability calculation
The probability is the ratio of favorable outcomes to the total number of possible outcomes. Thus, the probability (P) that exactly 2 minors will be selected is:
[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{60}{126} = \frac{10}{21} \approx 0.4762
]
Conclusion:
The probability that the witness will refuse to serve alcoholic beverages to exactly 2 minors, given that she randomly checks the IDs of 5 students from a group of 9 (with 4 minors and 5 legal students), is approximately 0.4762, or 47.62%.
This means that there is about a 47.62% chance that the witness will randomly select exactly 2 minors from the group of 5 students she checks.