Question

There are 10 girls and 15 boys in a race. What is the probability that the first, second and third place winners will be boys?

Answer 1

Given:

Number of girls = 10

Number of boys = 15

To find:

The probability that the first, second and third place winners will be boys.

Solution:

Total number of boys and girls:

$10+15=25$

The probability that the first place winner will be a boy is

$P(\text{Boy at first place})=\dfrac{15}{25}$

Now, the remaining number of boys is 14. So, the probability that the first place winner will be a boy is

$P(\text{Boy at second place})=\dfrac{14}{24}$

Now, the remaining number of boys is 13. So, the probability that the first place winner will be a boy is

$P(\text{Boy at third place})=\dfrac{13}{23}$

The probability that the first, second and third place winners will be boys is

$P(BBB)=\dfrac{15}{25}\times \dfrac{14}{24}\times \dfrac{13}{23}$

$P(BBB)=\dfrac{3}{5}\times \dfrac{7}{12}\times \dfrac{13}{23}$

$P(BBB)=\dfrac{1}{5}\times \dfrac{7}{4}\times \dfrac{13}{23}$

$P(BBB)=\dfrac{91}{460}$

Therefore, the probability that the first, second and third place winners will be boys is $\dfrac{91}{460}$.