Question

A and B play a game where each is asked to select a number from 1 to 25. If the numbers selected by A and B match, both of them win a prize. The probability that they win their third prize on $5^{th}$ game is equal to:

• A. $\frac{6.(24{)}^2}{(25{)}^5}$
• B. $\frac{6.(21{)}^2}{(25{)}^5}$
• C. $\frac{(24{)}^2}{(25{)}^2}$
• D. $\frac{6.(24{)}^2}{(25{)}^4}$

Answer 1

The correct option is A $\frac{6.(24{)}^2}{(25{)}^5}$

Probability of winning the prize in single game
= probability of getting {(1, 1), (2, 2), ………… (25, 25)}
$=\frac{25}{25\times 25}=\frac{1}{25}$
Probability(3rd prize in 5th game)
Probability (two prize in first 4 games) and prize in $5^{th}$ game
${=}^4{C}_2{\left(\frac{1}{25}\right)}^2{\left(\frac{24}{25}\right)}^2\times \frac{1}{25}=\frac{6\times (24{)}^2}{(25{)}^5}$