Question

Assuming that random guesses are made for ten multiple choice questions on an SAT, so there is n=10 trials, each with a probability of correct answers by p=0.2. That means each question has 5 choices with only 1 appropriate answer. Find the probability that the number if correct answers is exactly 6.

Answer 1

Okay, here we have this:

Here we are going to use the binomial probability equation to solve the problem, so we have the following formula:

Where:

n=total number of questions (trials)=10

k=number correct (successes)=6

n−k=number incorrect (failures)=4

p=probability of getting 1 question correct=(0.2)

q=1−p probability of getting 1 question incorrect=(0.8)

So we obtain the following:

$P=(\frac{n!}{k!(n-k)!})p^kq^{n-k}$

Replacing:

$\begin{gathered} P=(\frac{10!}{6!(10-6)!})0.2^6\cdot0.8^{10-4} \\ P=(\frac{10!}{6!(4!)})0.2^6\cdot0.8^6 \\ P=(\frac{10\cdot9\cdot8\cdot7}{4!})0.2^6\cdot0.8^6 \\ P=\frac{5040}{24}\cdot0.000064\cdot\: 0.262144 \\ P=\frac{0.08455716864}{24} \\ P=0.00352321536 \end{gathered}$