Question

What is the probability that Sue gets a B grade of at least 4 out of 5 on a 5-question multiple choice test (with answers A - D for each question) by guessing at random? What is the sample space? What are the events that satisfy the condition of at least 4 out of 5 correct answers?

Answer 1

Ci - Event of answering i^th question correct ,....i=1,2,3,4,5

Ii - Event of answering i_th Question incorrect ; i=1,2,3,4,5

Probability of answering a question correctly :p = 1/4 =0.25

Probability of answering a question incorrectly :q = 3/4 = 0.75

Let X : Number of questions answered correctly

1. Probability that Sue gets a B grade of at least 4 out of 5 on a 5-question multiple choice test i.e P(X4) = P(X=4)+ P(X=5)

Probability of answering all questions correctly :P(X=5)

P(X=5) = P(C1 and C2 and C3 and C4 and C5) = 0.25 x 0.25 x 0.25 x 0.25 = 0.25⁵

P(X=4) i.e answering 4 questions correctly and one question incorrectly there 5 choices for this event

choice 1 : 1st question incorrect i.e P(I1 and C2 and C3 and C4 and C5)=0.75x0.25 x 0.25 x 0.25x0.25 = 0.75 x 0.25⁴

choice 2 : 2nd question incorrect i.e P(C1 and I2 and C3 and C4 and C5)= 0.75x 0.25⁴

choice 3 : 3rd question incorrect i.e P(C1 and C2 and i3 and C4 and C5)= 0.75x 0.25⁴

choice 4: 4th question incorrect i.e P(C1 and C2 and C3 and I4 and C5)= 0.75x 0.25⁴

choice 5 : 5th question incorrect i.e P(C1 and C2 and C3 and C4 and I5)= 0.75x 0.25⁴

P(X=4) = 5 x 0.75 x 0.25⁴

P(X=4)+ P(X=5) = 0.25⁵ + 5 x 0.75 x 0.25⁴ = 0.015625

P(X4) = P(X=4)+ P(X=5) = 0.015625

Probability that Sue gets a B grade of at least 4 out of 5 on a 5-question multiple choice test = 0.015625

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2. The Sample Space

Sample space-

5 multiple answer questions, each question can be answered correctly(C) or incorrectly(I);

Number of samples = 2⁵ = 32;

Sample number	Number of correct answers	Event space
1	5	C1,C2,C3,C4,C5
2	4	C1,C2,C3,C4,I5
3	4	C1,C2,C3,I4,C5
4	4	C1,C2,I3,C4,C5
5	4	C1,I2,C3,C4,C5
6	4	I1,C2,C3,C4,C5
7	3	I1,I2,C3,C4,C5
8	3	I1,C2,I3,C4,C5
9	3	I1,C2,C3,I4,C5
10	3	I1,C2,C3,C4,I5
11	3	C1,I2,I3,C4,C5
12	3	C1,I2,C3,I4,C5
13	3	C1,I2,C3,C4,I5
14	3	C1,C2,I3,I4,C5
15	3	C1,C2,I3,C4,I5
16	3	C1,C2,C3,I4,I5
17	2	I1,I2,I3,C4,C5
18	2	I1,I2,C3,I4,C5
19	2	I1,I2,C3,C4,I5
20	2	I1,C2,I3,I4,C5
21	2	I1,C2,I3,C4,I5
22	2	I1,C2,C3,I4,I5
23	2	C1,I2,I3,I4,C5
24	2	C1,I2,I3,C4,I5
25	2	C1,I2,C3,I4,I5
26	2	C1,C2,I3,I4,I5
27	1	C1,I1,I2,I3,I4,I5
28	1	I1,C2,I3,I4,I5
29	1	I1,I2,C3,I4,I5
30	1	I1,I2,I3,C4,I5
31	1	I1,I2,I3,I4,C5
32	0	I1,I2,I3,I4,I5

The events that satisfy the condition of at least 4 out of 5 correct answers; i.e number of correct answers = 4 or 5

From the above event space, following are the event space that satisfy the condition of at least 4 out of 5 correct answers is

Event number	Number of correct answers	Event space
1	5	C1,C2,C3,C4,C5
2	4	C1,C2,C3,C4,I5
3	4	C1,C2,C3,I4,C5
4	4	C1,C2,I3,C4,C5
5	4	C1,I2,C3,C4,C5
6	4	I1,C2,C3,C4,C5