Answer to Question #185052 in Statistics and Probability for Irsya Balqis

Question

From a recent survey in Cambridge University, it was discovered that 8% of the students are short-sighted.

a) If 20 students are randomly selected, calculate :

i. the probability that none of them are short-sighted

ii. the probability that at most 2 are short-sighted

iii. the standard deviation for the number of short-sighted student

 

b) If 50 classes of 20 students are randomly selected, what is the probability that 10 classes have no short-sighted students?

Answer 1

p=0.08;q=1-0.08=0.92

a) n=20

Let x be a random variable denoting the number of short sighted students.

$P(X=x) ={n \choose x} p^xq^{n-x}$

i ) P(X=0)

$={20 \choose 0}*0.08^{0}*0.92^{20}$

=0.19

ii) $P(X \le2)$

$=\sum_{x=0}^2{20 \choose x} 0.08^x0.92^{20-x}$

=0.79

iii) standard deviation

$=\sqrt{npq}$

$=\sqrt {20*0.08*0.92}$

=1.21

b) the probability that ten classes have no short sighted students.

From (a) (i) the probability that there is no short sighted student is 0.19.

From 50 classes, the probability that 10 classes have no short sighted students is;

$={ 50\choose 10}* 0.19^{10*}0.81^{40}$

=0.14