Question

$A$ can choose a lottery ticket out of three in which there is only $1$ prize and $2$ blanks and $B$ can choose $3$ lottery tickets out of $9$ in which there are $3$ prizes and $6$ blanks. If winning at least one prize is a success, then the ratio of probability of $A$'s success to that of $B$'s success is

• A. $2:3$
• B. $3:8$
• C. $9:19$
• D. $7:16$

Answer 1

The correct option is D $7:16$

$A$ can draw a ticket in ${}^3{C}_1=3$ ways.
The number of cases in which $A$ can get a prize is clearly $1.$
$\therefore$ Probability of $A$'s success $=\frac{1}{3}$

Again, $B$ can draw a ticket in ${}^9{C}_3=\frac{9\cdot 8\cdot 7}{3\cdot 2\cdot 1}=84$ ways.
The number of ways in which $B$ gets all blanks $={}^6{C}_3=\frac{6\cdot 5\cdot 4}{3\cdot 2\cdot 1}=20$
$\therefore$ The number of ways of getting a prize $=84-20=64$
$\therefore$ Probability of $B$'s success $=\frac{64}{84}=\frac{16}{21}$

Hence, required ratio $=\frac{1}{3}:\frac{16}{21}=7:16$