Question

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number $1,2,3,.....12$ as shown in Fig. What is the probability that it will point to:
(i) $10$ (ii) an odd number (iii) a number which is a multiple of $3$ (iv) an even number.

Answer 1

Solution(i):

Let $E$ be event of pointing to $10$ in the game of chance with numbers $1$ to $12$

No. of favorable outcomes$=1$$(i.e.,10)$

Total no. of possible outcomes $=12$

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

Therefore, the probability that the arrow points to $10$ is $\frac{1}{12}$

Solution(ii):

Let $F$ be event of pointing to an odd number in the game of chance with numbers $1$ to $12$

Odd no. up to $12=1,3,5,7,9,11$
No. of favorable outcomes$=6$

Total no. of possible outcomes $=12$

We know that, Probability $P\left(F\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{6}{12}=\frac{1}{2}$

Therefore, the probability that the arrow points to an odd number is $\frac{1}{2}$

Solution(iii):

Let $H$ be event of pointing to a multiple of $3$ in the game of chance with numbers $1$ to $12$

Multiples of $3$ up to $12=3,6,9,12$
No. of favorable outcomes$=4$

Total no. of possible outcomes $=12$

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

Therefore, the probability that the arrow points to a multiple of $3$ is $\frac{1}{3}$

Solution(iv):

Let $D$ be event of pointing to an even number in the game of chance with numbers $1$ to $12$

Even no. up to $12=2,4,6,8,10,12$
No. of favorable outcomes$=6$

Total no. of possible outcomes $=12$

We know that, Probability $P\left(D\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{6}{12}=\frac{1}{2}$

Therefore, the probability that the arrow points to an even number is $\frac{1}{2}$