A bag contains six balls of different colours.

Question

A bag contains six balls of different colours. Two balls are drawn in succession with replacement; probability both same colour is $p$. Next, four balls are drawn in succession with replacement; probability exactly three are the same colour is $q$. If $p:q=m:n$ (coprime), find $m+n$.

Answer: 14

Accepted Answer

$$p = \frac{6\times1}{6\times6}=\frac16,\qquad q=\frac{6\times{}^5C_1\times4}{6^4}$$

Wait --- using the source's working: $q=\dfrac{6\cdot{}^5C_1\cdot4}{6^4}=\dfrac{5}{54}$.

$$p:q = \frac16:\frac{5}{54} = 9:5 \implies m+n=14$$

Solutions · 1