Question

A bag contains $14$ balls of two colours, the number of balls of each colour being the same. $7$ balls are drawn at random one by one. The ball in hand is returned to the bag before each new draw. If the probability that at least $3$ balls of each colour are drawn is $p$ then

• A. $p>\frac{1}{2}$
• B. $p=\frac{1}{2}$
• C. $p<1$
• D. $p<\frac{1}{2}$

Answer 1

The correct option is A $p>\frac{1}{2}$

There are two possible cases.
Case One, $3$ balls are drawn of first color and $4$ balls are drawn of second color.
Case Two, $4$ balls are drawn of first color and $3$ balls are drawn of second color.
The total probability is therefore,

${}^7{C}_3{\left(\frac{7}{14}\right)}^3{\left(\frac{7}{14}\right)}^4{+}^7{C}_4{\left(\frac{7}{14}\right)}^4{\left(\frac{7}{14}\right)}^3=\frac{35}{64}>\frac{1}{2}$