SOLUTION: An exam has ten true-false questions. A student who has not studied answers all ten questions by just guessing. Find the probability that the student correctly answers the given

Question 475934: An exam has ten true-false questions. A student who has
not studied answers all ten questions by just guessing. Find
the probability that the student correctly answers the given
number of questions.
(a) All ten questions
(b) Exactly seven questions
thankyou! :)
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
This is a binomial distribution problem. Recall that the pdf of a binomial distribution for n number of trials with probability p is

P(X = x) = (n C x)*(p)^x*(1-p)^(n-x)

For more help with binomial distributions, see this calculator.

In this case, n = 10 (ie there are 10 trials since there are 10 questions) and p=1/2=0.5

a)

In part a), x = 10 (since we want to get all 10 correct)

P(X = 10) = (10 C 10)*(0.5)^(10)*(1-0.5)^(10-10)

P(X = 10) = (10 C 10)*(0.5)^(10)*(0.5)^(10-10)

Note: 10 C 10 = (10!)/(10!(10-10)!) = 1

P(X = 10) = (1)*(0.5)^(10)*(0.5)^0

P(X = 10) = (1)*(0.0009765625)*(1)

P(X = 10) = 0.0009765625

So the probability of getting all 10 correct is 0.0009765625 (which is roughly 0.098 % .... a very very small chance)
----------------------------------------------------------

b)

In part b), x = 7 (since we want exactly 7)

P(X = 7) = (10 C 7)*(0.5)^(7)*(1-0.5)^(10-7)

P(X = 7) = (10 C 7)*(0.5)^(7)*(0.5)^(10-7)

Note: 10 C 7 = (10!)/(7!(10-7)!) = 120

P(X = 7) = (120)*(0.5)^(7)*(0.5)^3

P(X = 7) = (120)*(0.0078125)*(0.125)

P(X = 7) = 0.1171875

So the probability of getting exactly 7 correct is 0.1171875 (which is roughly 11.72%)

For more help with binomial distributions, see this calculator.
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