Question

In the High School there is around 2500 students, 18% of the students smoke cigarettes.

A) If 2 are selected at random, use a Venn diagram with 2 circles; 1 representing the probability that the first student smokes and 1 representing the probability that the other student smokes. Determine the probability that at least 1 of them smokes cigarettes. (This would be equivalent to the probability that either the first student OR the second student smokes.)

B) Repeat the above analysis when 3 students are selected at random.

Note: These trials would be independent given the large population of students.

Answer 1

Given, number of students in high school is around 2500 and 18% of them smoke cigarettes.

So, the number of students who smoke cigerattes is 18/100*2500 = 450

Hence ,by definition of probability, the probability of any chosen student smoking cigeratte ,

p = 450/2500 = 0.18

A.)  A random sample of 2 students is chosen.

The probability that atleast one of them smokes cigeratte =

Pr(any one of them smokes cigeratte) + Pr( both of them smoke cigeratte)   

Note that , total probability is always unity ( i.e. 1 ),

Pr( none of them smokes cigeratte ) +Pr(any one of them smokes cigeratte) + Pr( both of them smoke cigeratte) = 1

Pr(any one of them smokes cigeratte) + Pr( both of them smoke cigeratte)

= 1 - Pr( none of them smokes cigeratte )   ............ (*)

Now, probability that none of them smokes

= Pr(first student doesnot smoke) * Pr(second student doesnot smoke )

= 0.18 * 0.18 ( since chances of smoking of one student doesnot depend on the other )

= 0.0324  

Putting this value in ....(*),

The probability that atleast one of them smokes cigeratte = 1 - 0.0324 = 0.9676

B.) A random sample of 3 students is chosen.

Similarly, The probability that atleast one of them smokes cigeratte =

Pr(any one of them smokes cigeratte) + Pr( two of the three smoke cigeratte) + Pr( all 3 smoke )

Accordingly, just like before the total probability of unity is obtained as :

Pr( none of them smoke cigeratte )+Pr(any one of them smokes cigeratte) + Pr( two of the three smoke cigeratte) + Pr( all 3 smoke ) = 1

Pr(any one of them smokes cigeratte) + Pr( two of the three smoke cigeratte) + Pr( all 3 smoke )

= 1 - Pr( none of them smoke cigeratte ) .............. (**)

Now, Pr( none of them smoke )

=Pr(1st student doesnot smoke) * Pr(2nd student doesnot smoke) * Pr (3rd student doesnot smoke)

= 0.18 * 0.18 * 0.18 ( since choice of smoking is independent)

= 0.005832

Putting this value in ...(**),

The probability that atleast one of them smokes cigeratte = 1-0.005832 = 0.994168