Question

A school newspaper reporter decides to randomly survey 17 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 21% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

a. In words, define the Random Variable X.

b. List the values that X may take on.

c. Give the distribution of X.

X ~ ____ ( ____ , ____ )

d. How many of the 17 students do we expect to attend the festivities? (Round your answer to the nearest whole number.)   

e. Find the probability that at most 3 students will attend. (Round your answer to four decimal places.)

f.   Find the probability that more than 2 students will attend. (Round your answer to four decimal places.)

Answer 1

Ans a A random variable X is a variables whose values are numerical outcomes of a random phenomenon.It is of two types 1. Discrete and 2. Continuous

Ans b X here can be 0 meaning no student attend Tet to 17 that will mean every student attended Tet. It cannot be negative since it is count of students or any decimal value and neither it can be greater than 17 since total number of students is 17. So the values X can take is { 0 ,1 , 2 . . . .,17} .

Ans c The distribution of X is

X is Binomial(n= 17 and p = 0.21)

where n will be the sample size which is 17 here and p is the probability that a student will attend Tet which is 0.21 estimated from last year's attendance.Since it is a case of n similar objects independently having two options with same probability the random variable has to follow binomial in such condition.

Ans d Here we need to calculate E[X] which is expected value of X .Since X is a binomial variable and for a binomial variable the expected value is n*p which has been defined with its values in (c)

E[X] = 17 * 0.21 = 3.57

Ans e Before solving this question we should know that the probability distribution function for Binomial variable X is given by

P[X = k] = where k is one of the values X can take and n and p as defined in (c)

Required to calculate the probability that atmost 3 students attended the festival which is basically

P[X <= 3] = P[X=0] + P[X=1] + P[X=2] + P[X=3]

=                

= 1 * 1 * 1.821 * 10^-3 + 17 * 0.21 * 2.64 * 10^-3 + 136 * 0.0441 * 3.826* 10^-3 + 680 * 9.261*10^-3 * 5.545 * 10^-3 = 0.0691 approximately

Ans f In this question we need to find the probability that P[X>=2] which is equivalent to calculating 1 - P[X<2]

P[X<2] = P[X = 0] + P[X=1]

               = = 1 * 1 * 1.821 * 10^-3 + 17 * 0.21 * 2.64 * 10^{-3 =} 0.01124 approximately

Therefore P[X>=2] = 1 - P[X<2] = 1 - 0.01124 = 0.98875 = 0.9888 approximately