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:
[tex]10+15=25[/tex]
The probability that the first place winner will be a boy is
[tex]P(\text{Boy at first place})=\dfrac{15}{25}[/tex]
Now, the remaining number of boys is 14. So, the probability that the first place winner will be a boy is
[tex]P(\text{Boy at second place})=\dfrac{14}{24}[/tex]
Now, the remaining number of boys is 13. So, the probability that the first place winner will be a boy is
[tex]P(\text{Boy at third place})=\dfrac{13}{23}[/tex]
The probability that the first, second and third place winners will be boys is
[tex]P(BBB)=\dfrac{15}{25}\times \dfrac{14}{24}\times \dfrac{13}{23}[/tex]
[tex]P(BBB)=\dfrac{3}{5}\times \dfrac{7}{12}\times \dfrac{13}{23}[/tex]
[tex]P(BBB)=\dfrac{1}{5}\times \dfrac{7}{4}\times \dfrac{13}{23}[/tex]
[tex]P(BBB)=\dfrac{91}{460}[/tex]
Therefore, the probability that the first, second and third place winners will be boys is [tex]\dfrac{91}{460}[/tex].
