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June 26, 2022, 11:31:03 am

### AuthorTopic: Probability theory task.  (Read 615 times)

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#### Sullivan Jones

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##### Probability theory task.
« 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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##### Re: Probability theory task.
« 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 »

#### Sullivan Jones

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##### Re: Probability theory task.
« Reply #2 on: May 18, 2022, 09:13:26 pm »
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Thank you! I am very grateful for your help. The probability that R guesses exactly 8 of the questions correctly is obtained below:
From the given information, let the random variable X be the number of correct answer follows Binomial distribution with there are 10 questions are randomly selected.  At https://plainmath.net/35995/a-class-is-taking-a-multiple-choice-test-with-10-questions-where-each I read this information and I understood how to solve this problem for me. Thank you very much for your help.
« Last Edit: May 27, 2022, 04:20:21 pm by Sullivan Jones »