Question

a test has eight multiple choice questions, each with four answer choices. if she guesses on each of the questions, what is the probability that kara gets at least six of the questions correct?

Explanation:

Step1: Identify the distribution

This is a binomial probability problem. The binomial probability formula is $P(X = k) = \binom{n}{k}p^{k}(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial. Here, $n = 8$ (number of questions), $p=\frac{1}{4}$ (probability of guessing a question correctly, since there are 4 choices), and we need to find $P(X\geq6)=P(X = 6)+P(X = 7)+P(X = 8)$.

Step2: Calculate $P(X = 6)$

First, calculate the binomial coefficient $\binom{8}{6}=\frac{8!}{6!(8 - 6)!}=\frac{8\times7}{2\times1}=28$. Then, $p^{6}=(\frac{1}{4})^{6}=\frac{1}{4096}$, and $(1 - p)^{2}=(\frac{3}{4})^{2}=\frac{9}{16}$. So, $P(X = 6)=28\times\frac{1}{4096}\times\frac{9}{16}=\frac{28\times9}{4096\times16}=\frac{252}{65536}$.

Step3: Calculate $P(X = 7)$

The binomial coefficient $\binom{8}{7}=\frac{8!}{7!(8 - 7)!}=8$. Then, $p^{7}=(\frac{1}{4})^{7}=\frac{1}{16384}$, and $(1 - p)^{1}=\frac{3}{4}$. So, $P(X = 7)=8\times\frac{1}{16384}\times\frac{3}{4}=\frac{24}{65536}$.

Step4: Calculate $P(X = 8)$

The binomial coefficient $\binom{8}{8}=1$. Then, $p^{8}=(\frac{1}{4})^{8}=\frac{1}{65536}$, and $(1 - p)^{0}=1$. So, $P(X = 8)=1\times\frac{1}{65536}\times1=\frac{1}{65536}$.

Step5: Sum the probabilities

Now, sum $P(X = 6)$, $P(X = 7)$, and $P(X = 8)$:

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Answer:

The probability that Kara gets at least six questions correct is $\frac{277}{65536}\approx0.00423$ (or approximately $0.423\%$).