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Question
A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has at most one girl?
Solution
Total number of students = (10 + 8) = 18
Let S be the sample space.
Then n(S) = number of ways of selecting 3 students out of 18 = 18C3 ways
Let E be the event with at most one girl in the group.
Then E = {0 girl, 1 girl}
∴ Favourable number of events, n(E) = 8C0 × 10C3 × 8C1 × 10C2
Hence, the required probability is given by
\[\frac{^{8}{}{C}_0 \times^{10}{}{C}_3 + ^{8}{}{C}_1 \times ^{10}{}{C}_2}{^{18}{}{C}_3}\]
\[ = \frac{1 \times^{10}{}{C}_3 + ^{8}{}{C}_1 \times ^{10}{}{C}_2}{^{18}{}{C}_3}\]
\[ = \frac{1 \times 120 + 8 \times 45}{816}\]
\[ = \frac{480}{816}\]
\[ = \frac{10}{17}\]
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