Question

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

• A. A little less than $\frac{1}{2}$
• B. a little greater than $\frac{1}{2}$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A A little less than $\frac{1}{2}$

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) $=9C_5{\left(\frac{1}{2}\right)}^9$
Cases (ii)
P(getting 4 white, 5 black) $=9C_4{\left(\frac{1}{2}\right)}^9$
These are exclusive so
P(atleast 4 of each colour) $=9C_5{\left(\frac{1}{2}\right)}^9+9C_4{\left(\frac{1}{2}\right)}^9=\frac{63}{128}$ which is little less than $\frac{1}{2}$