Question

If the overall percentage of success in an exam is $60$ , what is the probability that out of a group of $4$ students, at least one has passed?
a) $0.6525$
b) $0.9744$
c) $0.8704$
d) $0.0256$

Answer 1

Here we have been given an observation and we have to find the probability of an event to occur. We will solve this problem by using Binomial Probability distribution. Firstly we will calculate the probability of success/pass and failure. Then we will use the binomial formula and simplify it to get the desired answer.

Complete step-by-step solution:
It is given to us that the overall percentage of success in an exam is $60$ .
Total percentage of success in an exam is $100$ .
So probability of pass is as follows:
Probability of pass $=\dfrac{60}{100}$
Probability of pass $=\dfrac{3}{5}$
We denote it as $p=0.6$
Next we will find the probability of failure,
Probability of failure $=1-p$
Probability of failure $=1-0.6$
Probability of failure $=0.4$
We denote it by $q=0.4$
Total student given is $=4$
We will denote it as $n=4$
Now as we have to find the probability that at-least one has passed it will be equal to total probability minus the probability of none-passed.
So probability that none passed will be as follows,
None-passed probability $={}^{4}{{C}_{n}}\times {{p}^{0}}\times {{q}^{4}}$
Put the value of $p=0.6$ , $q=0.4$ and $n=4$ use the formula of combination as follows:
None-passed probability $={}^{4}{{C}_{4}}\times {{\left( 0.6 \right)}^{0}}\times {{\left( 0.4 \right)}^{4}}$
None-passed probability $=\dfrac{4!}{4!\left( 4-4 \right)!}\times {{\left( 0.6 \right)}^{0}}\times {{\left( 0.4 \right)}^{4}}$
None-passed probability $=0.0256$
So probability that at-least one passed $=1-$ None-passed probability
Probability that at-least one passed $=1-0.0256$
Probability that at-least one passed $=0.9744$
Hence the correct option is (B).

Note: Binomial probability distribution is the discrete probability of the numbers of success in any sequence of independent experiments. The answer in which case is either yes or no there is no other outcome. In this case either the students failed or they passed was the only outcome that is why we used binomial probability distribution formula here.