Question

A bag contains 5 red marbles and 5 green marbles. One marble is drawn, its colour recorded, and then placed back into the bag. This process is repeated until a green marble is found. Given that the first green marble is found on an odd-numbered draw, the probability that it is found on the fifth draws, is

• A. $\frac{1}{32}$
• B. $\frac{3}{64}$
• C. $\frac{1}{256}$
• D. $\frac{3}{256}$

Answer 1

Answer: (B)

E: green marble is drawn on an odd numbers draw
$p(S)=\frac{1}{2};p(F)=\frac{1}{2}$
$p(E)=p(S$ or $FFS$ or $FFFFS$ or $\dots .)=.\frac{p(S)}{1-p(F)p(F)}=\frac{\frac{1}{2}}{1-\frac{1}{4}}=\frac{1}{2}\cdot \frac{4}{3}=\frac{2}{3}$
Let $A$ : Green marble is drawn on this $5^{\text{th }}$ draw
$\therefore p(A/E)=\frac{p(A\cap E)}{p(E)}=\frac{p(A)}{p(E)}=\frac{\frac{1}{32}}{\frac{2}{3}}=\frac{1}{32}\cdot \frac{3}{2}=\frac{3}{64}$