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 the probability of passing if the minimum passing...

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 the probability of passing if the minimum passing grade is 70 %

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

The minimum passing grade is `70%` means at least `7` questions must be answered correctly. At least `7` means exactly `7` or exactly `8` or exactly `9` or exactly `10.` All these possibilities are mutually incompatible, so we'll sum up their probabilities.

The probability of answering each questions correctly is `1/4,` incorrectly is `3/4.` Denote the number of `k`-combinations from `n` as `C_n^k.`

Consider answering correctly all `10` questions. The probability is `(1/4)^10` because correct or incorrect answer for the next question is independent from any previous and the probabilities of independent events are multiplied.

For exactly `9` correct answers the probability is `10*(3/4)*(1/4)^9` because one (but arbitrary) of `10` questions must be answered incorrectly (`10` possibilities) and `9` remaining correctly.

For exactly `8` it is `C_10^2*(3/4)^2*(1/4)^8,`  (there are `C_10^2` possibilities of choosing `2` questions to fail), for `7` it is  `C_10^3*(3/4)^3*(1/4)^7.`

The result may be expressed as   `sum_(k=0)^3 C_10^k*(3/4)^k*(1/4)^(10-k).`

 

In numbers it isn't so great, it is about 0.0035, or 0.35%.

vkinard | Student

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`

` `