What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly checked 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 need to find the probability that a witness will refuse to serve alcoholic beverages to exactly two minors when randomly checking the IDs of five students selected from a group of nine students, where four of them are minors (not of legal age).
Key Information:
- Total students: 9
- Minors: 4
- Legal age students: 5 (since there are 9 students in total)
- Number of students to be checked: 5
- We want the probability that exactly 2 minors are selected when the witness checks 5 students.
Step 1: Total number of ways to choose 5 students from 9
The total number of ways to select 5 students from 9 students is calculated using the combination formula:
[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
]
where ( n = 9 ) and ( k = 5 ). So, the total number of ways to choose 5 students from 9 is:
[
\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: Number of favorable outcomes (selecting exactly 2 minors)
To ensure that exactly 2 minors are selected, we need to choose 2 minors from the 4 minors, and the remaining 3 students must be selected from the 5 legal-age students. We use combinations again for this:
- The number of ways to select 2 minors from 4 minors:
[
\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
] - The number of ways to select 3 legal-age students from 5 legal-age students:
[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
]
Step 3: Total favorable outcomes
The total number of favorable outcomes (selecting exactly 2 minors and 3 legal-age students) is the product of the two combinations:
[
\binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60
]
Step 4: Probability
Finally, the probability is the ratio of favorable outcomes to the total number of outcomes. The probability is:
[
P(\text{exactly 2 minors}) = \frac{60}{126} = \frac{10}{21}
]
Final Answer:
The probability that the witness will refuse to serve alcoholic beverages to exactly two minors is 10/21 or approximately 0.476.
Explanation:
In this problem, we used the concept of combinations because the order of selection does not matter. We first calculated the total possible ways to select 5 students from 9, and then we calculated the number of ways to select exactly 2 minors and 3 legal-age students. The ratio of these two values gives the probability that exactly 2 minors will be selected when 5 students are randomly chosen. This is a classic example of probability with combinations, where we need to calculate outcomes with restrictions (i.e., selecting exactly 2 minors out of 4).