Question

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

Answer 1

Hint: Use permutations and combinations to get the number of favourable outcomes and the total number of outcomes. For solving the above question, you need to make two cases, one where the ball transferred from the first bag is white and the other that the ball drawn from the first bag is black.

Complete step-by-step answer:
Before moving to the question, let us talk about probability.
Probability, in simple words, is the possibility of an event to occur.
Probability can be mathematically defined as $=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .
Now, let’s move to the solution to the above question. So, to start with the solution, we will make two cases, one where the ball transferred from the first bag is white and the other that the ball drawn from the first bag is black.
So, let us solve for the first case, i.e., the ball taken from the first bag is white. The probability that the ball selected from the first bag is white is equal to $\dfrac{\text{number of white balls in first bag}}{\text{total number of balls in the first bag}}=\dfrac{4}{9}$ . Now, in this case, the total number of balls after adding the white ball picked from first bag is 14 and the probability of getting white from second bag becomes: $\dfrac{\text{number of white balls in second bag}}{\text{total number of balls in the second bag}}=\dfrac{7}{14}=\dfrac{1}{2}$ .
Now let us solve for the other case, i.e., the ball taken from the first bag is black. The probability that the ball selected from the first bag is black is equal to $\dfrac{\text{number of black balls in first bag}}{\text{total number of balls in the first bag}}=\dfrac{5}{9}$ . Now, in this case, the total number of balls after adding the black ball picked from first bag is 14 and the probability of getting white from second bag becomes: $\dfrac{\text{number of white balls in second bag}}{\text{total number of balls in the second bag}}=\dfrac{6}{14}=\dfrac{3}{7}$ .
So, the final probability that the ball drawn is white is:
$\begin{align}
  & \dfrac{\text{number of white balls in first bag}}{\text{total number of balls in the first bag}}\times \dfrac{\text{number of white balls in second bag}}{\text{total number of balls in the second bag}}+ \\
 & \dfrac{\text{number of black balls in first bag}}{\text{total number of balls in the first bag}}\times \dfrac{\text{number of white balls in second bag}}{\text{total number of balls in the second bag}} \\
\end{align}$
$=\dfrac{4}{9}\times \dfrac{1}{2}+\dfrac{5}{9}\times \dfrac{3}{7}=\dfrac{2}{9}+\dfrac{5}{21}=\dfrac{14+15}{63}=\dfrac{29}{63}$
Therefore, the answer to the above question is $\dfrac{29}{63}$ .

Note: It is preferred that while solving a question related to probability, always cross-check the possibilities, as there is a high chance you might miss some or have included some extra or repeated outcomes. Also, when a large number of outcomes are to be analysed then permutations and combinations play a very important role.