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, we need to determine the chance that a witness (like a bartender) randomly checks the IDs of 5 students chosen from a group of 9 students, where 4 of the 9 students are minors (underage), and she ends up identifying only two minors among the five she checks. This implies she checks five people and finds exactly two of them to be minors.
Step-by-step Solution:
We are choosing 5 students from 9 total (4 minors and 5 adults). We want exactly 2 of the 5 checked to be minors.
Let’s break this down:
- Total number of ways to choose 5 students from 9:
[
\binom{9}{5} = 126
] - Number of favorable outcomes (selecting exactly 2 minors from 4, and 3 adults from 5):
[
\binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60
] - Probability:
[
P = \frac{60}{126} = \frac{10}{21} \approx 0.4762
]
Final Answer: (\boxed{\frac{10}{21}})
Explanation (300 words):
In this scenario, a witness such as a bartender is responsible for checking IDs to prevent minors from being served alcohol. There are 9 students in total: 4 are minors (not of legal age) and 5 are adults. The bartender checks the IDs of 5 randomly selected students. We are asked to find the probability that exactly two of those five are minors.
This is a classic hypergeometric probability problem because we are drawing a sample from a finite population without replacement, and we are interested in the exact number of “successes” (minors in this case) in the sample.
To find this probability, we use combinations:
- First, determine the total number of ways to choose 5 students from 9. This is given by (\binom{9}{5} = 126).
- Next, we calculate how many of those combinations include exactly 2 minors. That means:
- Choose 2 minors from the 4 available: (\binom{4}{2} = 6)
- Choose 3 adults from the 5 available: (\binom{5}{3} = 10)
- Total favorable combinations = (6 \times 10 = 60)
- So, the probability is the ratio of favorable outcomes to total outcomes: (\frac{60}{126} = \frac{10}{21})
This result means there’s roughly a 47.6% chance that the bartender checks IDs of five students and ends up identifying exactly two minors — which would result in denying alcohol to just those two. Understanding such probabilities helps in assessing how effective random ID checks are in identifying underage drinkers.
