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  • 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 t

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

Option: 1

\mathrm{p>\frac{1}{2}}


Option: 2

\mathrm{p=\frac{1}{2}}


Option: 3

\mathrm{\frac{90}{2^{10}}}


Option: 4

None of these


Answers (1)

best_answer

A ball drawn is of either colour. If drawing a ball of one colour is a success then drawing ball of the other colour is a failure.

\therefore  the probability of success in one draw =\frac{7}{14}=\frac{1}{2}  and the probability of failure in one draw =\frac{7}{14}=\frac{1}{2}

\therefore  the probability that at least 3 balls of each colour is drawn

\begin{aligned} & = \mathrm{{ }^7 C_3 \cdot\left(\frac{1}{2}\right)^3 \cdot\left(\frac{1}{2}\right)^4+{ }^7 C_4\left(\frac{1}{2}\right)^4 \cdot\left(\frac{1}{2}\right)^3=2 \cdot \frac{7 !}{(3 !)(4 !)} \cdot \frac{1}{2^7}} \\ \\& =2 \cdot \frac{7 \cdot 6 \cdot 5}{6} \cdot \frac{1}{2^7}=\frac{35}{64}>\frac{1}{2} \end{aligned}

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