Question

A fair die is rolled. The probability that the first time 1 occurs at the even throw, is
A. $\dfrac{1}{6}$
B. $\dfrac{5}{11}$
C. $\dfrac{6}{{11}}$
D. $\dfrac{5}{{36}}$

Answer 1

Hint: Here, we will be using the concept of probability to solve the question. The probability of an event is the chance that the event occurs. Odd numbers are the natural numbers which are not divisible by 2. First, we will find the total number of outcomes when a die is rolled. We will then find the favourable outcomes of getting a 1. We will substitute the values in the formula of probability to find the answer.

Formula Used: We will use the formula for the probability of an event
$P(E)=Number of favorable outcomes/total outcomes$
The sum of favorable outcomes and unfavorable outcomes is equal to 1

Complete step by step Solution:
The extent to which something is likely to happen is basically what probability means. This is the fundamental theory of probability, which is also applied to the probability distribution, and from which you will discover the likelihood of results in a random experiment.
First, we will find the total number of outcomes and the number of favorable outcomes.
When a die is rolled, the possible outcomes are 1, 2, 3, 4, 5, or 6 appearing on the die. Thus, we observe that the total number of outcomes is 6 when a die is rolled.

Let E be the event of getting 1 when a fair die is rolled.
Therefore we know that
$ \Rightarrow P(E) = \dfrac{1}{6}$ so, $P(\bar E) = \dfrac{5}{6}$
The probability that the first time 1 occurs at an even throw is given by:
$ = P(E)P(\bar E) + P(E)P(\bar E)P(\bar E)P(\bar E) + P(E)P(\bar E)P(\bar E)P(\bar E)P(\bar E).........\infty $
$ = (\dfrac{5}{6})(\dfrac{1}{6}) + {(\dfrac{5}{6})^3}(\dfrac{1}{6}) + {(\dfrac{5}{6})^5}(\dfrac{1}{6}) + ......\infty $
This becomes an infinite Geometric Progression (GP). So we can use the formula of infinite GP
$ = \dfrac{{\dfrac{5}{{36}}}}{{1 - \dfrac{{25}}{{36}}}} = \dfrac{5}{{11}}$

Hence, the correct option is B.

Note: Whenever we come up with this type of problem then first, we had to find the total number of possible outcomes (like here total possible numbers when die is rolled) and after that we had to find the number of favourable outcomes (like here number of times 1 occurs) and then we can directly apply the probability formula to find the probability of getting a favourable outcome. Then we can apply the formula to find the probability that first time any number (in our case its 1) occurs at an even throw