A box has 5 black and 3 green shirts. One shirt is picked randomly and put in another box. The second box has 3 black and 5 green shirts. Now a shirt is picked from second box. What is the probability of it being a black shirt?

Solution(By Examveda Team)

From box 1 we can pick black or green shirt
Case 1: Pick black shirt
Box 1 has total 5 + 3 = 8 shirts
Probability of black from box 1 = $$\frac{{5}}{{8}}$$
Now this black is added to box 2
So box 2 now has 3 + 1 = 4 black & 5 green shirts
Total = 4 + 5 = 9 shirts
Probability of black from box 2 = $$\frac{{4}}{{9}}$$
Case 1 probability
$$\eqalign{ & = \frac{5}{8} \times \frac{4}{9} \cr & = \frac{{20}}{{72}} \cr} $$

Case 2: Pick green shirt
Probability of green from box 1 = $$\frac{{3}}{{8}}$$
Now this green is added to box 2
So box 2 now has 3 black & 5 + 1 = 6 green shirts
Total = 3 + 6 = 9 shirts
Probability of black from box 2 = $$\frac{{3}}{{9}}$$
Case 2 probability
$$\eqalign{ & = \frac{3}{8} \times \frac{3}{9} \cr & = \frac{9}{{72}} \cr} $$
Total Probability
$$\eqalign{ & = \frac{{20}}{{72}} + \frac{9}{{72}} \cr & = \frac{{29}}{{72}} \cr} $$