Guessing on a multiple choice test is a binomial experiment. The success can be thought of as guessing the right answer to a question, and failure is guessing the wrong answer. The probability of success $p$ is equal to $1$ divided by number of questions — in this case, $3 .$ Similarly, the probability of failure is $q = \frac{2}{3} .$

Event	Probability
Success: Guessed Right Answer	$p = \frac{1}{3}$
Failure: Guessed Wrong Answer	$q = \frac{2}{3}$

The probability that we will guess at least

4

out of

5

questions right is equal to the sum of the probabilities that we will guess exactly

4

right answers and

5

right answers.

P (At least 4 right) = P (4 right) + P (5 right)

Let's calculate the probabilities one at a time, starting with

P (4 right) .

Probability of Four Correct Answers

To find the probability that we will guess $4$ right answers on a $5 -$ question long test, we can use the Binomial Probability Formula.

Binomial Probability
Suppose you have $n$ repeated independent trials each with a probability of success $p$ and a probability of failure $q$ (with $p + q = 1$ ). Then the binomial probability of $x$ successes in the $n$ trials can be found by the following formula. $P (x) =_{n} C_{x} p^{x} q^{n - x}$

Exercise

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