Question

A die is thrown $6$ times. If 'getting an odd number' is a success, then the probability of at most $5$ successes is:

• A. $\frac{1}{64}$
• B. $\frac{63}{64}$
• C. $\frac{1}{2}$
• D. $\frac{1}{2^5}$

Answer 1

The correct option is B $\frac{63}{64}$

We know that the repeated tosses of a dice are known as Bernoulli trials.
Let the number of successes of getting an odd number in an experiment of 6 trials be x.
Probability of getting an odd number in a single throw of a dice
$p=\frac{\text{number of odd numbers on a dice}}{\text{total number of numbers on a dice}}=\frac{3}{6}=\frac{1}{2}$
Thus, $q=1-p=\frac{1}{2}$
Now, here $X$ has a binomial distribution.
Thus, $P(X=x){=}^n{C}_x{q}^{n-x}{p}^x,$ where $x=0,1,2,...n$
${=}^6{C}_x{\left(\frac{1}{2}\right)}^{6-x}{\left(\frac{1}{2}\right)}^x{=}^6{C}_x{\left(\frac{1}{2}\right)}^6$
Probability of getting at most 5 successes $=P(X\le 5)$
We can also write it as: $1-P(X>5)$
$=1-P(X=6)=1{-}^6{C}_6{\left(\frac{1}{2}\right)}^6=1-\frac{1}{64}=\frac{63}{64}$