Question

On a 7 question multiple-choice test, where each question has 2 answers, what would be the probability of getting at least one question wrong?Give your answer as a fraction

Answer 1

- This is a Binomial probability question. The formula for Binomial probability is:

`\begin{gathered} P(r)=\sum\text{ }^nC_rp^rq^(n-r) \\ where, \\ n=The\text{ total number of trials} \\ r=\text{ The number of successful trials\lparen where answer is correct\rparen} \\ p=\text{ The probability of success \lparen The probability of getting a question } \\ right) \end{gathered}`

- We have been given:

`\begin{gathered} n=7 \\ \text{ since there can only be two answers, it means that the} \\ \text{ probability of getting a question correct is:} \\ p=(1)/(2) \\ q=1-p=(1)/(2) \\ \\ \text{ The probability of getting at least 1 question wrong means the } \\ probability\text{ of getting 1, 2, 3, 4, 5, 6, or 7 question wrong.} \\ \\ \text{ Instead of calculating all these probabilities, we can simply say} \\ P(1)+P(2)+P(3)+P(4)+P(5)+P(6)+P(7)=1-P(0) \end{gathered}`

- Thus, we have:

`\begin{gathered} P(0)=^7C_0((1)/(2))^0((1)/(2))^7 \\ P(0)=(1)/(128) \\ \\ 1-P(0)=1-(1)/(128)=(127)/(128) \end{gathered}`

Final Answer

The answer is

`(127)/(128)`