Question

A bag contains $3$ red balls, $5$ black balls and $4$ white balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is:
(i) white (ii) red (iii) black (iv) not red

Answer 1

Total no. of balls in the bag is $12$ $(3$ red, $5$ black and $4$ white$)$

Solution(i):

No. of white balls in bag is $4$

Therefore, ${}^4{C}_1$( Selecting $1$ out of $4$ items) times out of ${}^{12}{C}_1$( Selecting $1$ out of $11$ items) a white ball is picked.

Let $E$ be the event of drawing a white ball from bag

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^4{C}_1}{{}^{12}{C}_1}$$=\frac{4}{12}$$=\frac{1}{3}$

Solution(ii):

No. of red balls in bag is $3$

Therefore, ${}^3{C}_1$( Selecting $1$ out of $3$ items) times out of ${}^{12}{C}_1$( Selecting $1$ out of $12$ items) a red ball is picked.

Let $E$ be the event of drawing a red ball from bag

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^3{C}_1}{{}^{12}{C}_1}$$=\frac{3}{12}$$=\frac{1}{4}$

Solution(iii):

No. of black balls in bag is $5$

Therefore, ${}^5{C}_1$( Selecting $1$ out of $5$ items) times out of ${}^{12}{C}_1$( Selecting $1$ out of $12$ items) a black ball is picked.

Let $E$ be the event of drawing a black ball from bag

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^5{C}_1}{{}^{12}{C}_1}$$=\frac{5}{12}$

Solution(iv):

No. of balls that are not redin bag is $9$ $(5$ black, $4$ white$)$

Therefore, ${}^9{C}_1$( Selecting $1$ out of $9$ items) times out of ${}^{12}{C}_1$( Selecting $1$ out of $12$ items) a ball that is not red is picked.

Let $E$ be the event of drawing a ball that is not red from bag

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^9{C}_1}{{}^{12}{C}_1}$$=\frac{9}{12}$$=\frac{3}{4}$