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In a class of 20 students, there are 12 girls and 8 boys. Three students are randomly selected to be teacher's assistants.
-What is the probability that the group has at least two boys? Round your answer to four decimal places.
A) 0.3439
B) .04421
C) .02947
D) 0.5895

Question

Quizplus · Accepted Answer

The probability of selecting at least two boys can be found by calculating the probability of selecting exactly two boys and one girl, and the probability of selecting three boys, then adding these probabilities together. First, calculate the probability of selecting exactly two boys and one girl:- The number of ways to choose 2 boys from 8 is $ \binom{8}{2} $.- The number of ways to choose 1 girl from 12 is $ \binom{12}{1} $.- The total number of ways to choose 3 students from 20 is $ \binom{20}{3} $.So, the probability of choosing exactly two boys and one girl is:$$ P(2B,1G) = \frac{\binom{8}{2} \cdot \binom{12}{1}}{\binom{20}{3}} $$Next, calculate the probability of selecting three boys:- The number of ways to choose 3 boys from 8 is $ \binom{8}{3} $.- The total number of ways to choose 3 students from 20 remains $ \binom{20}{3} $.So, the probability of choosing three boys is:$$ P(3B) = \frac{\binom{8}{3}}{\binom{20}{3}} $$Adding these probabilities gives the total probability of selecting at least two boys:$$ P(\text{at least 2 boys}) = P(2B,1G) + P(3B) $$Calculating these probabilities and adding them together gives a result of approximately 0.3439, which matches choice A.

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