Question

The ideal size of the class for the students enrolled in a course is 150 at particular study centre. It is known from an earlier data that on the average only 30% of those candidates will actually attend the class suppose the study centre admits 450 students. What is the probability that more than 150 students attend the class for the course.

Answer 1

Now let X be the random variable indicating the number of students present.

Now it is given that the probability of a student being present is 30% so,

Now X follows binomial distribution with parameters $\left(n,\:p\right)=\left(450,\:0.3\right)$

Now as n approaches infinity and p is finite so we use binomial approximation to normal distribution.

$np=450\times 0.3=135;$

$\sqrt{\left(np\left(1-p\right)\right)}=\sqrt{\left(135\times 0.7\right)}$

We are required to find,

$\:P\left(X>150\right)=P\left(X>150.5\right)\:\left[for\:discrete\:to\:continuous\:approximation\right]$

$=P\frac{\left(Z>\left(150.5-135\right)\right)}{\sqrt{\left(135\times 0.7\right)}}$

$=P\left(Z>1.594\right)$

$=1-P\left(Z<1.59\right)$

$=1-\left(0.5+0.441\right)$

$=0.059$