Question

you will be assessed on the use of BINOM.DIST function, and creating formula to calculate the expected value of a binomial distribution. You will also be asked to interpret a probability value.

Multiple Choice Test

Chris is sitting a multiple choice test in statistics. The test consists of 9 items and each item has 5 options. After the test, Chris told you he didn’t study for the test at all and guessed all answers.

a)

If Chris guessed each and every item in the test, describe the probability distribution for the number of correct guesses. Present the probability distribution in the form of a table using 6 decimal points in your answers. (Use BINOM.DIST function to calculate the probabilities.)

Answer 1

Solution

Let X = Number of correct guesses. Then, X ~ B(n, p), where

n = number of items in the test = 9 (given)

p = probability of one correct guess = 1/5 [Because, guessing with 5 options could be any one of the options, but only one option is correct and hence the probability is 1/5.]

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)

[The above probability can also be directly obtained using Excel Function of Binomial Distribution: BINOMDIST(Number_s:Trials:Probability_s:Cumulative), what is within brackets is (x:n:p:False/True)] …………………………………………………………….(1a)

Now, to work out the solution,

Since there are 9 items in the test, X can take any value from 0 to 9, both inclusive.

The probabilities are given below:

x	p(x)
	by (1)	by (1a)
0	9C0(0.2)0(0.8)9	0.134218
1	9C1(0.2)1(0.8)8	0.30199
2	9C2(0.2)2(0.8)7	0.30199
3	9C3(0.2)3(0.8)6	0.176161
4	9C4(0.2)4(0.8)5	0.06606
5	9C5(0.2)5(0.8)4	0.016515
6	9C6(0.2)6(0.8)3	0.002753
7	9C7(0.2)7(0.8)2	0.000295
8	9C8(0.2)8(0.8)1	1.84E-05
9	9C9(0.2)9(0.8)0	5.12E-07
Total		1

DONE