A quiz consists of 10 multiple-choice questions, each with 4 possible answers. For someone who makes random guesses for all of the answers, find...

This is a great question, and one that you will probably see several times in a Probability and Statistics class.

The main thing you will need is the Binomial Probability formula, shown below:

`P(k)=nCk*p^k*(1-p)^(n-k)`

The first part is “n choose k”, where n is the total number of questions and k is the number of questions you want to consider.  So you are starting by finding all the possible combinations of choosing 7 random questions out of the 10.  That is then multiplied by the probability of getting a question right 7 times and by the probability of getting a question wrong 3 times.

So, to start, let’s consider the probability you need to find.  You want to find the probability of all possibilities of passing the quiz, which includes scores of 70%, 80%, 90%, and 100% totaled.  This is where you will need the Binomial Probability formula.

For our problem, n is the number of questions, 10, and k is the number of correct answers, 7 through 10.  The probability p of getting a question right is  and the probability of getting a question wrong is  or  . 

So, the probability of getting a score of exactly 70% is `10C7*(1/4)^7*(3/4)^(3)`

The probability of getting a score of exactly 80% is `10C8*(1/4)^8*(3/4)^(2)`

The probability of getting a score of exactly 90% is `10C9*(1/4)^9*(3/4)^(1)`

The probability of getting a score of exactly 100% is `10C10*(1/4)^10*(3/4)^(0)`

We now just add up all these probabilities for our answer:

`P(>= 70%) = 10C7*(1/4)^7*(3/4)^(3) + 10C8*(1/4)^8*(3/4)^(2) +10C9*(1/4)^9*(3/4)^(1) + 10C10*(1/4)^10*(3/4)^(0)`

``If you type this CAREFULLY into your graphing calculator, your final answer is:

`919/262144~~0.0035057`

` `