Answer:
[tex]P[at\ least\ 1] = 0.9961[/tex]
Step-by-step explanation:
Given
[tex]Questions = 8[/tex]
[tex]Quiz\ Type = True\ or\ False[/tex]
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:
[tex]P(correct) = \frac{1}{2}[/tex]
Similarly, the probability of answering a question, wrongly is:
[tex]P(wrong) = \frac{1}{2}[/tex]
The following relationship exists, in probability:
[tex]P[at\ least\ 1] = 1 - P[none][/tex]
So, to calculate the required probability.
First, we calculate the probability that he answers none of the 8 questions correctly.
[tex]P[none] = p(wrong)^8[/tex]
[tex]P[none] = (\frac{1}{2})^8[/tex]
Substitute [tex]P[none] = (\frac{1}{2})^8[/tex] in [tex]P[at\ least\ 1] = 1 - P[none][/tex]
[tex]P[at\ least\ 1] = 1 - (\frac{1}{2})^8[/tex]
[tex]P[at\ least\ 1] = 1 - \frac{1}{256}[/tex]
Take LCM
[tex]P[at\ least\ 1] = \frac{256 - 1}{256}[/tex]
[tex]P[at\ least\ 1] = \frac{255}{256}[/tex]
[tex]P[at\ least\ 1] = 0.9961[/tex]
Hence, the probability that s/he gets at least one correctly is 0.9961
