Question

A bag contains $4$ white and $3$ black balls. Another bag contains $5$ white and $2$ black balls. A pair of dice is rolled. If the sum on the dice is $10$, $1^{st}$ bag is selected. Otherwise $2^{nd}$ bag is selected. Find the probability of drawing white ball if one ball is drawn from the selected bag at random.

Answer 1

Sum can be $10$ if

$(4,6)(5,5)(6,4)$

Probability to get $10$ sum is $\frac{1}{6}\times \frac{1}{6}\times 3=\frac{1}{12}$

Probability to get sum other than $10=1-\frac{1}{12}=\frac{11}{12}$

Probability of white from first bag $=\frac{4}{7}$

Probability of white from second bag $=\frac{5}{7}$

Probability of getting white $=\frac{1}{12}\times \frac{4}{7}+\frac{11}{12}\times \frac{5}{7}=\frac{59}{84}$