Question

Peppermint Patty is very discouraged about her chances on a 10-item true-false quiz. If she randomly answers each question, what is her probability of getting a grade of at least 50% on the following? (Enter an exact number as an integer, fraction, or decimal.)
three-item quiz

four-item quiz

five-item quiz

Answer 1

We can solve this problem using the binomial probability equation:

P = [n! / (n – r)! r!] p^r q^(n – r)

where the variables are:

n = total number of questions = 10

r = number of correct = at least 5

p = probability of success = 0.5

q = probability of failure = 0.5

 

So what we have to do is to calculate for P for r = 5 to 10

 

when r = 5

P =  [10! / (10 – 5)! 5!] 0.5^5 0.5^(10 – 5)

P = 0.246

 

when r = 6

P =  [10! / (10 – 6)! 6!] 0.5^6 0.5^(10 – 6)

P = 0.205

 

when r = 7

P =  [10! / (10 – 7)! 7!] 0.5^7 0.5^(10 – 7)

P = 0.117

 

when r = 8

P =  [10! / (10 – 8)! 8!] 0.5^8 0.5^(10 – 8)

P = 0.044

 

when r = 9

P =  [10! / (10 – 9)! 9!] 0.5^9 0.5^(10 – 9)

P = 9.766 x 10^-3

 

when r = 10

P =  [10! / (10 – 10)! 10!] 0.5^10 0.5^(10 – 10)

P = 9.766 x 10^-4

 

So the probability that her score will be at least 5 is:

P (r≥5) = 0.246 + 0.205 + 0.117 + 0.044 + 9.766 x 10^-3 + 9.766 x 10^-4

P (r≥5) = 0.623

 

So about 62.3% chance.


You can do the same for the other item quiz, just set the value of n