Question

Consider two sets $P=\{a,b,c,d,e\}$ and $Q=\{f,g,h,i,j\}A$ person selects a set $P$ with probability $1/3$ or set $Q$ with probability $2/3$ and forms a subset $R$ of two elements both selected from set $P$ or $Q$. Find probability that set $R$ consists of one vowel $\&$ one consonants

• A. $\frac{1}{5}$
• B. $\frac{3}{11}$
• C. $\frac{2}{15}$
• D. $\frac{7}{15}$

Answer 1

Answer: (D)

$P(R)=P(P)\cdot P\left(\frac{V.C}{P}\right)+P(Q)\cdot P\left(\frac{V.C}{Q}\right)$
$=\frac{1}{3}\times \frac{{}^2{C}_1\cdot {}^3{C}_1}{{}^5{C}_2}+\frac{2}{3}\cdot \frac{{}^1{C}_1{}^4{C}_1}{{}^5{C}_2}$
$=\frac{1}{3}\times \frac{6}{10}+\frac{2}{3}\times \frac{4}{10}=\frac{6+8}{30}=\frac{7}{15}$