Name: Multiplication Theorem on Probability
Uploaded: 2019-07-19
Duration: 2 min 37 s
Description: Everything that happens in life is based on probability. Do you know what the multiplication theorem of probability is? Watch the video for a clear explanati...

Question

Three boxes numbered $I, I I, I I I$ contain the balls as follows:

$\begin{matrix} White & Black & Red \\ I & 1 & 2 & 3 \\ II & 2 & 1 & 1 \\ III & 4 & 5 & 3 \end{matrix}$

One box is randomly selected and a ball is drawn from it. If the ball is red, then find the probability that it is from Box- $I I$ .

Accepted Answer

Given: Three boxes numbered I, II, III contain the balls as follows:

White Black Red

I $1$ $2$ $3$

II $2$ $1$ $1$

III $4$ $I I$ $3$

Let $B_{1}, B_{2}, B_{3}$ be the events of selecting balls from first, second and third box respectively.

We know that, from the classic definition of probability $P (A) = \frac{n (A)}{n (S)}$ , where $n (A)$ is the number of outcomes favourable to $A$ and $n (S)$ is number of total outcomes.

Now, $P (B_{1}) = P (B_{2}) = P (B_{3}) = \frac{1}{3}$

Probability of selecting a red ball from the first box $= \frac{3}{6} = P (\frac{R}{B_{1}})$

Probability of selecting a red ball from the second box $= \frac{1}{4} = P (\frac{R}{B_{2}})$

Probability of selecting a red ball from the third box $= \frac{3}{12} = P (\frac{R}{B_{3}})$

Suppose the ball is red, probability it is from box II is found as follows: $P (\frac{B_{2}}{R}) = \frac{P (B_{2}) \cdot P (\frac{R}{B_{2}})}{P (B_{1}) \cdot P (\frac{R}{B_{1}}) + P (B_{2}) \cdot P (\frac{R}{B_{2}}) + P (B_{3}) \cdot P (\frac{R}{B_{3}})}$

$= \frac{\frac{1}{3} \cdot \frac{1}{4}}{\frac{1}{3} (\frac{3}{6} + \frac{1}{4} + \frac{3}{12})} = \frac{\frac{1}{4}}{\frac{1}{2} + \frac{1}{4} + \frac{1}{4}} = \frac{1}{4}$ .

Therefore, the required probability is $\frac{1}{4}$ .

Important Questions on Probability

Learn and Prepare for any exam you want !

Three boxes numbered I,II,III contain the balls as follows: White Black Red I 123 II 211 III 453 One box is randomly selected and a ball is drawn from it. If the ball is red, then find the probability that it is from Box-II.

Important Questions on Probability

Learn and Prepare for any exam you want !