Question

On an eight question true-false quiz, a student guesses each answer. What is the probability that he/she gets at least one of the answers correct?

Answer 1

Answer:

$P[at\ least\ 1] = 0.9961$

Step-by-step explanation:

Given

$Questions = 8$

$Quiz\ Type = True\ or\ False$

Required

Probability that s/he gets at least one correctly

First, we calculate the probability of answering a question correctly

Since, there are just 2 choices (true or false), the probability is:

$P(correct) = \frac{1}{2}$

Similarly, the probability of answering a question, wrongly is:

$P(wrong) = \frac{1}{2}$

The following relationship exists, in probability:

$P[at\ least\ 1] = 1 - P[none]$

So, to calculate the required probability.

First, we calculate the probability that he answers none of the 8 questions correctly.

$P[none] = p(wrong)^8$

$P[none] = (\frac{1}{2})^8$

Substitute $P[none] = (\frac{1}{2})^8$ in $P[at\ least\ 1] = 1 - P[none]$

$P[at\ least\ 1] = 1 - (\frac{1}{2})^8$

$P[at\ least\ 1] = 1 - \frac{1}{256}$

Take LCM

$P[at\ least\ 1] = \frac{256 - 1}{256}$

$P[at\ least\ 1] = \frac{255}{256}$

$P[at\ least\ 1] = 0.9961$

Hence, the probability that s/he gets at least one correctly is 0.9961