Question

a student takes a multiple choice test that has 30 questions. each question has 5 choices. the student did not study for the test and decides to randomly guess at each question. a) find the probability the student gets exactly 10 problems correct. b) calculate the mean. c) calculate the standard deviation. statistics and probability.

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ………….........................................................................………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST]....................…..........……….(1a)

Mean (average) of X = E(X) = µ = np….....................................................................……………………………………………..(2)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ……......................................................…………………………………...(2a)

Now, to work out the solution,

Let X = number of problems out of 30 that the student gets correct. Then, X ~ B(30, 0.2) ..................................................... (3)

Since the student guesses randomly, each of the 5 choices coming correct has equal probability of 1/5 = 0.2

Part (a)

Probability the student gets exactly 10 problems correct

= P(X = 10)

= (³⁰C₁₀)(0.2¹⁰)(0.8)²⁰ [vide (1) and (3)]

= 0.0355 [vide (1a)]  Answer 1

Part (b)

Vide (2),

Mean = 30 x 0.2

= 6 Answer 2

Part (c)

Vide (2a),

Standard Deviation = √(30 x 0.2 x 0.8)

= 2.19 Answer 3

DONE