Question

Two letters are taken at random from the word "HOME".

• A. The probability that both the letters are vowels is 1/6
• B. The probability that at least one is vowel is 5/6
• C. The probability that one of the letter is M is 1/2
• D. The probability that at least one is vowel is 1/6

Answer 1

Answer: (A), (B), (C)

The correct options are
A The probability that both the letters are vowels is 1/6
B The probability that at least one is vowel is 5/6
C The probability that one of the letter is M is 1/2

(A) Total number of letters in the given word $=$ 4, out of these O and E are two vowels.
Let S be the sample space, then
$S=\{HO,HM,HE,OM,OE,ME\}\Rightarrow n\left(S\right)=6$
Let $A_1$ be the event of taking two vowels, then
$A_1=\left\{OE\right\}=\frac{n\left(A_1\right)}{n\left(S\right)}=\frac{1}{6}$
(B) Let $A_2$ be the event of taking at least one vowel, then we have
$A_2=\{HO,HE,OE,ME,MO\}\Rightarrow n\left(A_2\right)=5$
Hence, required probability $=\frac{\left(n\right(A_2\left)\right)}{n\left(S\right)}=\frac{5}{6}$
(C) Let $A_3$ be the event of taking one letter as M, then we have $A_3=\{HM,OM,ME\}\Rightarrow n\left(A_2\right)=5$
Hence,required probability $=\frac{n\left(A_3\right)}{n\left(S\right)}=\frac{3}{6}=\frac{1}{2}$