What is the probability that Robert guesses exactly 8 of the questions correctly? (Round to four decimal places.)

What is the probability that Robert gets an

or better? Another way of saying this is, what is the probability that he guesses 8 or more questions correctly? This would be the probability of 8 plus the probability of 9 plus the probability of 10. (Round to four decimal places.)

Let \(X\) be the binomial random variable representing this situation where \(X\sim \text{Bi}\left(10,\frac{1}{4}\right)\), ie. we have 10 trials and the probability of success for each trial is \(\frac14\) as there are 4 possible options of each.

We want to find \(P\left(X=8\right)\) (probability that we get exactly 8 questions correctly) and we get this by \(\begin{pmatrix}10\\ 8\end{pmatrix}\left(\frac{1}{4}\right)^8\left(\frac{3}{4}\right)^2\). We multiply the probability of success (we have \(8\)) with the probability of failures (with have 2 failures) and we multiply this to the number of ways our successes and failures can be arranged (which is \(\begin{pmatrix}10\\ 8\end{pmatrix}\) or \(^{10}C_8\)

Can you use this to determine \(P\left(X\ge 8\right)\)

Hint: \(P\left(X\ge 8\right)=P\left(X=8\right)+P\left(X=9\right)+P\left(X=10\right)\)