Question

A dice is thrown \[5\] times then the probability that an even number will come up exactly \[3\] times is
1.\[\dfrac{5}{16}\]
2.\[\dfrac{1}{2}\]
3.\[\dfrac{3}{16}\]
4.\[\dfrac{3}{2}\]

Answer 1

Hint: In order to find out the probability that an even number will come up exactly \[3\] times when a dice is thrown \[5\] times, firstly we will be assigning the values to the variables by considering all the information given. Then we will be applying the concept of binomial distribution and solving it accordingly. This would be our required answer.

Complete step by step answer:
Now let us learn about binomial distribution. Binomial distribution is nothing but a discrete probability distribution that gives only two possible results in an experiment i.e. either success or loss. The formula of binomial distribution is \[P\left( x:n,p \right){{=}^{n}}{{C}_{x}}{{p}^{x}}{{\left( q \right)}^{n-x}}\]. Negative binomial distribution also exists. Binomial distribution can be calculated for mean and variance. The probability of success or failure varies for each trial.
Now let us start solving the given problem.
We are given that a dice is thrown \[5\] times.
Now let us assign the values to the variables in order to apply the concept of binomial distribution.
\[\begin{align}
  & p=\dfrac{1}{2} \\
 & q=\dfrac{1}{2} \\
 & n=5 \\
 & x=3 \\
\end{align}\]
Now let us solve it by applying the formula \[P\left( x:n,p \right){{=}^{n}}{{C}_{x}}{{p}^{x}}{{\left( q \right)}^{n-x}}\]
We get,
\[\begin{align}
  & P\left( x:n,p \right){{=}^{n}}{{C}_{x}}{{p}^{x}}{{\left( q \right)}^{n-x}} \\
 & {{\Rightarrow }^{5}}{{C}_{3}}{{\left( \dfrac{1}{2} \right)}^{3}}{{\left( \dfrac{1}{2} \right)}^{5-3}} \\
\end{align}\]
On further solving it, we get
\[\begin{align}
  & {{\Rightarrow }^{5}}{{C}_{3}}{{\left( \dfrac{1}{2} \right)}^{3}}{{\left( \dfrac{1}{2} \right)}^{5-3}} \\
 & {{\Rightarrow }^{5}}{{C}_{3}}\left( \dfrac{1}{8} \right)\left( \dfrac{1}{4} \right)=\dfrac{5!}{3!2!}\left( \dfrac{1}{8} \right)\left( \dfrac{1}{4} \right)=\dfrac{10}{32}=\dfrac{5}{16} \\
\end{align}\]
So when a dice is thrown \[5\] times then the probability that an even number will come up exactly \[3\] times is \[\dfrac{5}{16}\].

So, the correct answer is “Option 1”.

Note: While solving such problems, we must apply the appropriate concept for easy simplification. We must know clearly, if we have to apply permutation or combination because not following the correct method is the commonly committed error.