From a bag containing 9 distinct white and 9 distinct black, 9 balls are d

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From a bag containing 9 distinct white and 9 distinct black, 9 balls are drawn at random one by one, the drawn balls being replaced each time. The probability that at least four balls of each colour is in the draw, is

Options:

A . A little less than 12

B . a little greater than 12

C . 12

D . 13

Answer: Option A
:
A
Out of 9 distinct black and 9 distinct white balls, probability of drawing a white ball

= \frac{1}{2}

and drawing black ball is also

\frac{1}{2} .

for at least 4 of each colour in 9 draws with replacement, there are two cases,
Case(i)
P(getting 5 white, 4 black)

= 9 C_{5} {(\frac{1}{2})}^{9}

Cases (ii)
P(getting 4 white, 5 black)

= 9 C_{4} {(\frac{1}{2})}^{9}

These are exclusive so
P(atleast 4 of each colour)

= 9 C_{5} {(\frac{1}{2})}^{9} + 9 C_{4} {(\frac{1}{2})}^{9} = \frac{63}{128}

which is little less than

\frac{1}{2}

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