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 problem, we will use probability theory and the concept of combinations. The goal is to find the probability that a witness will refuse to serve alcoholic beverages to exactly two minors, given that she randomly checks the IDs of five students from a group of nine students, where four are minors and five are of legal age.
Given Data:
- Total number of students = 9
- Number of minors = 4
- Number of students of legal age = 5
- Number of students whose IDs will be checked = 5
We want to calculate the probability that exactly two of the five students whose IDs are checked are minors. This means we want the witness to select exactly two minors from the four minors, and the remaining three students must be of legal age (from the five students of legal age).
Step 1: Total Possible Outcomes
The total number of ways to select 5 students from 9 students is given by the combination formula:
[
\text{Total possible outcomes} = \binom{9}{5} = \frac{9!}{5!(9 – 5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
]
Step 2: Favorable Outcomes
To have exactly two minors among the five students, the witness must choose 2 minors from the 4 minors, and 3 students of legal age from the 5 students of legal age. This can be calculated as:
[
\text{Ways to choose 2 minors} = \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
]
[
\text{Ways to choose 3 students of legal age} = \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
]
Thus, the total number of favorable outcomes is:
[
\text{Favorable outcomes} = \binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60
]
Step 3: Probability Calculation
Now, we can calculate the probability of the witness refusing to serve alcoholic beverages to exactly two minors by dividing the number of favorable outcomes by the total possible outcomes:
[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} = \frac{60}{126} = \frac{10}{21}
]
Final Answer:
The probability that the witness will refuse to serve alcoholic beverages to exactly two minors is ( \frac{10}{21} ), which is approximately 0.476 or 47.6%.
Explanation:
In this problem, we are using the concept of combinations to calculate the number of ways to select a subset of students from a larger group. By focusing on how to select exactly two minors and three legal-age students, we calculated the number of favorable outcomes. Dividing this by the total number of possible ways to select 5 students from the group of 9 gives us the required probability.