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June 26, 2022, 02:45:17 pm

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#### Sullivan Jones

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« on: May 15, 2022, 10:14:01 pm »
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Hi there!
I will be very grateful if you can help me and explain in detail the solution to this task.

A class is taking a multiple choice test with 10 questions where each question has four possible answers. Assume that the answer to any question is independent of that to any other answer. Robert has forgotten to study for this test, so he simply guesses for each question.
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.)

#### 1729

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« Reply #1 on: May 16, 2022, 11:46:53 am »
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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: $P\left(X\ge 8\right)=P\left(X=8\right)+P\left(X=9\right)+P\left(X=10\right)$
« Last Edit: May 16, 2022, 12:37:19 pm by 1729 »

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