Question

Mrs. Hudson plans to give a 10 question multiple choice exam in which each question has 4 options. Each question counts for 1 point. A score of 8 or more is a passing score while 7 or less is a failing score. The probabilities of guessing a certain number of questions correctly are listed in the table below.

Number of Questions Correct
Probability for Guessing Correctly
0
0.05631351
1
0.18771172
2
0.28156757
3
0.25028229
4
0.14599800
5
0.05839920
6
0.01622200
7
0.00308990
8
0.00038624
9
0.00002861
10
0.00000095
Compute the probability of answering 8 or more questions correctly by calculating the probability with a direct sum and by using complements.
How do the two approaches compare?

Answer 1

0.0004158

0.00041581

Explanation:

Given that :

Number of Questions Correct

Probability for Guessing Correctly

0

0.05631351

1

0.18771172

2

0.28156757

3

0.25028229

4

0.14599800

5

0.05839920

6

0.01622200

7

0.00308990

8

0.00038624

9

0.00002861

10

0.00000095

Probability of answering 8 or more questions correctly ;

Direct sum:

P(x ≥ 8) = p(8) + p(9) + p(10)

P(≥ 8) = 0.00038624 + 0.00002861 + 0.00000095

= 0.0004158

Complement :

P(x ≥ 8) = 1 - p(x ≤ 7)

1 - [p(7) + p(6) + p(5) + p(4) + p(3) + p(2) + p(1) + p(0)]

1 - [0.00308990 + 0.01622200 + 0.05839920 + 0.14599800 + 0.25028229 + 0.28156757 + 0.18771172 + 0.05631351]

1 - 0.99958419

= 0.00041581