Since the student is guaranteed seven correct responses, she needs at least three of the remaining questions to be correct. Thus we need the probability that she gets 3,4, or 5 of the remaining questions correct.
This situation is an example of a binomial probability distribution. Each event can be described in a binary fashion (either correct or incorrect), each event is independent (the results of one does not effect the others), the probabilities do not change, and there are a finite number of events.
The formula for a binomial probability is given by:
where k is the number of successes we seek, n is the number of trials, and p is the probability of success. (For example, suppose we want the probability of 3 successes. The probability of the 1st success is 1/4, the next 1/4, and the 3rd 1/4. The remaining events are failures with a probability of 3/4 each. Using the multiplication principle, we get `(1/4)^3 * (3/4)^2` . However, this can occur in 10 different ways. Suppose we number the 5 questions 1–5. Then she could give correct answers to questions 123, 124, 125, 134, 135, 145, 234, 235, 245, or 345.)
Here n=5, p=1/4, and we want k=3,4, or 5.
`P(x=4) = ._5C_4(1/4)^4(3/4)=5(1/256)(3/4)=15/1024 `
`P(x=5) = ._5C_5(1/4)^5(3/4)^0=1/1024`
Since these are mutually exclusive, the probability of the three events is the sum of the probabilities: P(x=3,4, or 5)=106/1024=53/512 or about 10%.