Question

A die is thrown 2n+1 times. The probability of getting 1 or 3 or 4 at most n times is
(a) $\dfrac{1}{2}$
(b) $\dfrac{1}{3}$
(c) $\dfrac{1}{4}$
(d) None of these

Answer 1

Hint:Before we proceed with the problem, it is important to know about the term binomial distribution. It is a simple formula-based question on the binomial distribution.First, we will calculate the probability of getting 1 or 3 or 4. And further substituting the value in the binomial distribution formula we get our required result.

Formula Used: Required probability \[ = {}^{2n + 1}{C_0}{p^0}{q^{2n + 1}} + {}^{2n + 1}{C_1}{p^1}{q^{2n + 1 - 1}} + {}^{2n + 1}{C_2}{p^2}{q^{2n + 1 - 2}} + ....... + {}^{2n + 1}{C_n}{p^n}{q^{2n + 1 - n}}\]

Complete step by step Solution:
The binomial distribution is a probability distribution that models the probability of obtaining one of two outcomes either Success or Failure under a given number of parameters. It summarizes the no. of trials when each trial has the same chance of attaining one specific outcome. The value of a binomial is obtained by multiplying the number of independent trials by the successes.
In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the Boolean-valued outcome is represented either with success (probability p) or failure (probability q = 1 − p).
A single success/failure test is also called a Bernoulli trial. For n = 1, i.e., a single experiment, the binomial distribution is a Bernoulli distribution.
Let p be the probability of getting 1 or 3 or 4
$ \Rightarrow p = 0.5$ and $q = 1 - p = 0.5$
By using binomial distribution, the required probability = P (0 times) + P (1 times) + P (2 times) + ……. +P (n times)
\[ = {}^{2n + 1}{C_0}{p^0}{q^{2n + 1}} + {}^{2n + 1}{C_1}{p^1}{q^{2n + 1 - 1}} + {}^{2n + 1}{C_2}{p^2}{q^{2n + 1 - 2}} + ....... + {}^{2n + 1}{C_n}{p^n}{q^{2n + 1 - n}}\]
Hence by solving the above equation after putting values of p and q we get,
Net probability$ = \dfrac{1}{2}$

Hence, the correct option is A.

Note: For using the binomial distribution, the number of trials in an experiment should be fixed or finite. Each trial is independent on its own. This means none of the trials should have effect on the probability of the next trial. Each trial will have an equal probability of occurrence. The probability of success should be exactly the same from one trial to another.