Classroom

Of the 26 pupils in the classroom, 12 boys and 14 girls, four representatives are picked to the odds of being:
a) all the girls
b) three girls and one boy
c) there will be at least two boys

Correct answer:

p₁ =  0.067
p₂ =  0.2922
p₃ =  0.6409

Step-by-step explanation:

$p_1=\frac{14\cdot 13\cdot 12\cdot 11}{26\cdot 25\cdot 24\cdot 23}=\frac{77}{1150}=0.067$

$$\begin{gathered} C_3(14)=\left(\genfrac{}{}{0ex}{}{14}{3}\right)=\frac{14!}{3!(14-3)!}=\frac{14\cdot 13\cdot 12}{3\cdot 2\cdot 1}=364 \\ {C}_1(12)=\left(\genfrac{}{}{0ex}{}{12}{1}\right)=\frac{12!}{1!(12-1)!}=\frac{12}{1}=12 \\ {C}_4(26)=\left(\genfrac{}{}{0ex}{}{26}{4}\right)=\frac{26!}{4!(26-4)!}=\frac{26\cdot 25\cdot 24\cdot 23}{4\cdot 3\cdot 2\cdot 1}=14950 \\ {p}_2=\left(\genfrac{}{}{0ex}{}{14}{3}\right)\cdot \left(\genfrac{}{}{0ex}{}{12}{1}\right)/\left(\genfrac{}{}{0ex}{}{26}{4}\right)=364\cdot 12/14950=0.2922 \end{gathered}$$

$p_3=1-p_1-p_2=1-0.067-0.2922=0.6409$

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