Please help with the following statistics question Suppose y

Please help with the following statistics question,

Suppose you are taking an exam with 10 questions and you are required to get 7 or more right answers to pass. You go into the exam without studying so you are guessing at each question. With a 10-question true/false, what is the probability of passing? With a 10-question, multiple-choice test (six possible choices) where there is only one correct choice for each question, what is the probability of passing? Which test would be hardest to pass by guessing? Explain

Solution

a)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.5      
x = our critical value of successes =    7      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   6   ) =    0.828125
          
Thus, the probability of at least   7   successes is  
          
P(at least   7   ) =    0.171875 [answer]

**********************

b)

Here,

p = probability of right answer = 1/6 = 0.166666667

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.166666667      
x = our critical value of successes =    7      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   6   ) =    0.999732479
          
Thus, the probability of at least   7   successes is  
          
P(at least   7   ) =    0.000267521 [answer]

**********************

c)

The multiple choice test is harder by guessing, as the probability of getting a right answer is 1/6 = 0.1666667, compared to a true or false question in which p = 0.5.