Compute the probability of answering 8 or more questions correctly by calculating the probability with a direct sum and by using complements.

Question

I  did most of the work but I need some help on the last few questions! Thank you in advance!

 

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

I. 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

There are only 3 ways to get 8 or more correct, which are to get 8 correct, 9 correct, or 10 correct.  

 

So the direct sum of P(8 or more correct) = P(8) + P(9) + P(10)  

 

The complement approach is to find the probability of the complement event (which is getting 7 or fewer correct) and then subtracting this probability from 1, which is given by  

 

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

 

If you do both calculations you should find they are the same to within the round-off errors of the probabilities in your table.  You can decide for yourself which calculation is easier for this case.