Question

A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls. One bag is selected at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is red.

Answer 1

Hint: In this question, firstly we find the probability of selecting one bag and selecting the second bag. Then we find the probability of the red ball in one bag. After that, we find the probability of the red ball in the second bag. We use the formula of conditional probability to get the desired probability of ball drawn.

Complete step by step answer:
A red ball can be drawn in two mutually exclusive ways
Selecting bag I and then drawing a red ball from it.
Selecting bag II and then drawing a red ball from it.
Now, we consider the event (E1) as the event for selecting bag I and event (E2) as the event for selecting bag II.
E1 = selecting bag I
E2 = selecting bag II
Since one of the bags is selected randomly, the probability of both the event are:
$\begin{align}
  & P\left( {{E}_{1}} \right)=\dfrac{1}{2} \\
 & P({{E}_{2}})=\dfrac{1}{2} \\
\end{align}$
The probability of drawing a red ball when the first bag has been selected is denoted as $P\left( \dfrac{A}{{{E}_{1}}} \right)$.
Since, there are a total 7 balls amongst which 4 balls are red.
$P\left( \dfrac{A}{{{E}_{1}}} \right)=\dfrac{4}{7}$.
The probability of drawing a red ball when the second bag has been selected is denoted as $P\left( \dfrac{A}{{{E}_{2}}} \right)$.
Now, there are 6 balls in total and 2 red balls.
$P\left( \dfrac{A}{{{E}_{2}}} \right)=\dfrac{2}{6}$.
Using the law of total probability, we have
P (red ball) $=P\left( A \right)=P\left( {{E}_{1}} \right)P\left( \dfrac{A}{{{E}_{1}}} \right)+P\left( {{E}_{2}} \right)P\left( \dfrac{A}{{{E}_{2}}} \right)$.
$P\left( A \right)=\dfrac{1}{2}\times \dfrac{4}{7}+\dfrac{1}{2}\times \dfrac{2}{6}$
$\begin{align}
  & P\left( A \right)=\dfrac{2}{7}+\dfrac{1}{6} \\
 & P\left( A \right)=\dfrac{12+7}{42} \\
 & P\left( A \right)=\dfrac{19}{42} \\
\end{align}$
Hence, the probability of a red ball drawn from one of the two bags is represented as $P\left( A \right)\text{ is }\dfrac{19}{42}$.

Note: The key step in this problem is using the appropriate method of probability. For such questions, we always employ conditional probability first and then use the law of total probability. In this way, no mistake is committed and the final result is obtained. Calculation must be performed carefully to avoid errors.