What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly check If 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 calculate the probability that a witness will refuse to serve alcoholic beverages to exactly two minors when they randomly check the IDs of five students from a group of nine students, where four are minors (under legal age) and five are of legal age.
Step 1: Understanding the Setup
- Total number of students: 9
- Number of minors (students under legal age): 4
- Number of students of legal age: 5
- Number of students the witness checks: 5
We want to find the probability that exactly 2 minors are chosen when 5 students are selected. This is a hypergeometric distribution problem, where we are sampling without replacement.
Step 2: Define the Hypergeometric Distribution
The hypergeometric distribution describes the probability of getting exactly ( k ) successes (minors) in ( n ) draws (checks), where the population contains ( K ) successes (minors) and ( N-K ) failures (students of legal age).
The formula for the hypergeometric probability is:
[
P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}
]
Where:
- ( K ) is the total number of minors (4 minors),
- ( N ) is the total number of students (9 students),
- ( n ) is the number of students checked (5 students),
- ( k ) is the number of minors we want to choose (2 minors).
Step 3: Calculate the Probability
We will calculate the probability of choosing exactly 2 minors out of the 5 students checked. Using the formula:
- ( \binom{4}{2} ): This is the number of ways to choose 2 minors from the 4 minors.
- ( \binom{5}{3} ): This is the number of ways to choose 3 students of legal age from the 5 students of legal age.
- ( \binom{9}{5} ): This is the total number of ways to choose 5 students from the 9 students.
Now, let’s compute these combinations:
[
\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
]
[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
]
[
\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1} = 126
]
Now, applying the values in the hypergeometric probability formula:
[
P(X = 2) = \frac{\binom{4}{2} \binom{5}{3}}{\binom{9}{5}} = \frac{6 \times 10}{126} = \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} ), or approximately 0.4762 (47.62%).
Explanation:
This probability calculation is based on the idea of choosing two minors and three students of legal age out of a group of nine students. We applied the hypergeometric distribution formula because the sampling is without replacement (the IDs of students are checked without putting them back). The result ( \frac{10}{21} ) represents the likelihood of this specific outcome occurring in this scenario.