Question

If an individual is taking a 20 question multiple choice test, and they guess on each question, making the probability of getting any one question correct is 25%.  

What is the expected number of questions out of those 20 that an individual gets correct, and what is the variance?

What is the probability of getting exactly 8 questions correct?

What is the probability of getting at least one question correct i.e.

Answer 1

Answer)

As there are fixed number of trials and probability of each and every trial is same and independent of each other

Here we need to use the binomial formula

P(r) = ncr*(p^r)*(1-p)^n-r

Ncr = n!/(r!*(n-r)!)

N! = N*n-1*n-2*n-3*n-4*n-5........till 1

For example 5! = 5*4*3*2*1

Special case is 0! = 1

P = probability of single trial = 0.25

N = number of trials = 20

R = desired success

A)

Expected value = n*p = 20*0.25 = 5

B)

P(8) = 20c8*(0.25^8)*(1-0.25)^20-8 = 0.06088668922

C)

P(at least 1) = 1 - p(0)

As the sum of the probabilities is = 1

= 0.996828788