ISI2015-MMA-52

Question

Two coins are tossed independently where $P$(head occurs when coin $i$ is tossed) $=p_i, \: i=1,2$. Given that at least one head has occurred, the probability that coins produced different outcomes is

• A. $\frac{2p_1p_2}{p_1+p_2-2p_1p_2}$
• B. $\frac{p_1+p_2-2p_1p_2}{p_1+p_2-p_1p_2}$
• C. $\frac{2}{3}$
• D. none of the above

Answer 1

$\underline{\mathbf {Answer:}}$ $\mathbf B$

$\underline{\mathbf{Explanation:}}$

This question is a combination of both conditional and total probability.

P(different outcomes | given at least one head has occurred)

$=\large{\frac{\text {P(different outcome)}\cap \mathrm P\text {(at least one head)}}{\text {P(at least one head)}} = \mathrm{\frac{p_1(1-p_2)+p_2(1-p_1)}{p_1(1-p_2)+p_2(1-p_1)+p_1p_2}}}$

which can be further simplified to

$=\mathrm{ \Large{\frac{(p_1+p_2) - 2p_1p_2}{(p_1+p_2)-p_1p_2}}}$

$\therefore \mathbf B$ is the correct option.