Question

A student guesses the answers to 6 questions on a true-false quiz. Find the probability that the indicated number of guesses are correct:

exactly 4

Answer 1

four out of 6, so four right, two wrong

Answer 2

the probability she gets a right answer is the same as the probability she gets a wrong answer, they are both \(\frac{1}{2}\) so the probability she gets the particular sequence
right, right, right, right, wrong, wrong
is \(\frac{1}{2}\times\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=(\frac{1}{2})^6\)

Answer 3

the question is how many such sequences are there, or how many ways can we pick 4 right out of 6 wrong, or how many ways can we pick4 from a set of 6
the answer is called "four choose six" written in notation as \(\dbinom{6}{4}\) and computed as
\[\dbinom{6}{4}=\frac{6\times 5}{2}=3\times 5=15\]

Answer 4

so the probability she gets exactly 4 right out of 6 is \[\frac{15}{2^6}\]

Answer 5

oh jeez....i'll try to remember that..

Answer 6

you can use pascal's triangle to compute these for small values if you like