Question

Bag B₁​ contains 4 white and 2 black balls. Bag B₂​ contains 3 white and 4 black balls. A bag is drawn at random and a ball is chosen at random from it. What is the probability that the ball drawn is white?

Answer 1

Hint:First find the probability that the first bag is selected. Then find the probability that the ball drawn is white. After that multiply both to get the probability that the white ball is drawn from bag B₁. Now, find the probability that the second bag is selected. Then find the probability that the ball drawn is white. After that multiply both to get the probability that the white ball is drawn from bag B₂. Now add both the probability to get the probability that the white ball is drawn.

Complete step by step answer:
Given:- Bag B₁​ contains 4 white and 2 black balls.
Bag B₂​ contains 3 white and 4 black balls.
The probability of selecting bag B₁ is
$ \Rightarrow P\left( {{B_1}} \right) = \dfrac{1}{2}$
Now, the probability of drawing a white ball from bag B₁ is,
$P\left( {{B_1}W} \right) = P\left( {{B_1}} \right) \times P\left( W \right)$
Substitute the values,
$ \Rightarrow P\left( {{B_1}W} \right) = \dfrac{1}{2} \times \dfrac{4}{6}$
Cancel out the common factors,
$ \Rightarrow P\left( {{B_1}W} \right) = \dfrac{1}{3}$ ….. (1)
The probability of selecting bag B₂ is
$ \Rightarrow P\left( {{B_2}} \right) = \dfrac{1}{2}$
Now, the probability of drawing a white ball from bag B₂ is,
$P\left( {{B_2}W} \right) = P\left( {{B_2}} \right) \times P\left( W \right)$
Substitute the values,
$ \Rightarrow P\left( {{B_2}W} \right) = \dfrac{1}{2} \times \dfrac{3}{7}$
Multiply the terms,
$ \Rightarrow P\left( {{B_2}W} \right) = \dfrac{3}{{14}}$ ….. (2)
Therefore, the probability of drawing the ball as white is
$P\left( W \right) = P\left( {{B_1}W} \right) \times P\left( {{B_2}W} \right)$
Substitute the values from the equation (1) and (2)
$ \Rightarrow P\left( W \right) = \dfrac{1}{3} + \dfrac{3}{{14}}$
Take LCM on the right side,
$ \Rightarrow P\left( W \right) = \dfrac{{14 + 9}}{{42}}$
Add the terms in the numerator,
$\therefore P\left( W \right) = \dfrac{{23}}{{42}}$

Hence, the probability that the white ball drawn is $\dfrac{{23}}{{42}}$.

Note: Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
$P\left( E \right)=\dfrac{n\left( A \right)}{n\left( S \right)}$
A probability of 0 means that an event is impossible.
A probability of 1 means that an event is certain.
An event with a higher probability is more likely to occur.
Probabilities are always between 0 and 1.