Given:
The number of black balls = 8.
The number of red balls = 8.
Five balls are drawn without replacement.
Required:
We need to find the probability that exactly 3 blackballs are drawn.
Explanation:
The total number of balls in an urn =8+8 = 16 balls.
The total number of selections of 5 balls of which three are black balls are drawn.
[tex]=3\text{ black balls and 2 red balls}[/tex][tex]=8C_3\times8C_2[/tex][tex]=\frac{8!}{3!(8-3)!}\times\frac{8!}{2!(8-2)!}[/tex]
[tex]=\frac{8!}{3!\times5!}\times\frac{8!}{2!\times6!}[/tex][tex]=\frac{8\times7\times6\times5!}{3\times2\times5!}\times\frac{8\times7\times6!}{2\times6!}[/tex][tex]=8\times7\times4\times7[/tex][tex]=1568[/tex]
All possible out for drawing 5 balls is
[tex]=16C_5[/tex][tex]=\frac{16!}{5!(16-5)!}[/tex][tex]=\frac{16\times15\times14\times13\times12\times11!}{5\times4\times3\times2\times11!}[/tex][tex]=16\times3\times7\times13[/tex][tex]=4368[/tex]
The probability that exactly 3 blackballs are drawn is
[tex]P(exactly\text{ 3\rparen=}\frac{1568}{4368}[/tex]
[tex]P(exactly\text{ 3\rparen=0.3590}[/tex]
Final answer:
[tex]P(exactly\text{ 3\rparen=0.3590}[/tex]
