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There are eight black balls and eight red balls in an urn. If 5 balls are drawn without replacement what is the probability that exactly 3 blackballs are drawn express your answer as a fraction or decimal number rounded to four decimal places

There Are Eight Black Balls And Eight Red Balls In An Urn If 5 Balls Are Drawn Without Replacement What Is The Probability That Exactly 3 Blackballs Are Drawn E class=

Sagot :

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]

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