# Multiple choice test : random variable

• MHB
• mathmari
In summary, the conversation is about finding the probability distribution for a multiple choice test with 10 questions and 5 possible answers for each question. The random variable $X$ represents the number of correct answers out of 10, with a probability of $\frac{1}{5}$ for each correct answer and $\frac{4}{5}$ for each wrong answer. The probability distribution for this scenario is a binomial distribution. In part (b), the probabilities for different outcomes are calculated, including the probability of exactly 4 questions being answered correctly, more than 4 questions being answered correctly, all questions being answered correctly, at least half of the questions being answered correctly, and at least 5 and at most 8 questions being answered
mathmari
Gold Member
MHB
Hey!

A multiple choice test consists of 10 questions. For every question there are five possible answers, of which exactly one is correct. A test candidate answers all questions by chance.
(a) Give a suitable random variable with value range and probability distribution in order to work on part (b) with it.
(b) Determine (with intermediate steps) the probability that
(i) exactly 4 questions were answered correctly,
(ii) more than 4 questions have been answered correctly,
(iv) at least half of the questions were answered correctly,
(v) at least 5 and at most 8 questions have been answered correctly.For (a) :
Let $X$ be a random variable that describes the number of correct answers out of $10$, right?
For each correct answer the probability is equal to $\frac{1}{5}$ and each wrong answer has the probability $\frac{4}{5}$.

Is that correct so far? :unsure:

mathmari said:
For (a) :
Let $X$ be a random variable that describes the number of correct answers out of $10$, right?
For each correct answer the probability is equal to $\frac{1}{5}$ and each wrong answer has the probability $\frac{4}{5}$.
Hey mathmari!

Yep. (Nod)

We still need to identify the probability distribution function for (a) though.

Klaas van Aarsen said:
Yep. (Nod)

So is the range of the random variable $\{0,1,2,3,4,5,6,7,8,9,10\}$ ? :unsure:
Klaas van Aarsen said:
We still need to identify the probability distribution function for (a) though.

Do we have to calculate for each value of the random variable the corresponding probability? :unsure:

mathmari said:
So is the range of the random variable $\{0,1,2,3,4,5,6,7,8,9,10\}$ ?

Yep. (Nod)
mathmari said:
Do we have to calculate for each value of the random variable the corresponding probability?

It think we should just identify the name of the probability distribution and its parameters.
If we want to, we can also calculate the corresponding probabilities of the possible outcomes.

Klaas van Aarsen said:
It think we should just identify the name of the probability distribution and its parameters.

What do you mean by the name of probability distribution an its parameters? Is it $P(X=k)$ with $k\in\{0, 1, 2, \ldots , 10\}$? :unsure:
Klaas van Aarsen said:
If we want to, we can also calculate the corresponding probabilities of the possible outcomes.

This is then part (b), or not? :unsure:

mathmari said:
What do you mean by the name of probability distribution an its parameters? Is it $P(X=k)$ with $k\in\{0, 1, 2, \ldots , 10\}$?

You wrote:
mathmari said:
For each correct answer the probability is equal to $\frac{1}{5}$ and each wrong answer has the probability $\frac{4}{5}$.
This describes a Bernoulli distribution with parameter $p=\frac 15$, which is for an experiment with a single yes-no question.
However, we don't have a single yes-no question, but we have 10 questions.
We are looking for a distribution for the number of successes in a sequence of $n$ independent experiments, each asking a yesâ€“no question, and each with its own Boolean-valued outcome: success (with probability $p$) or failure (with probability $q = 1 âˆ’ p$). (Sweating)

mathmari said:
This is then part (b), or not?
More or less. We can use a table with the probabilities of each possible outcome to help us answer each of the questions in (b).
But we can also find the answers for (b) in a different fashion.

Klaas van Aarsen said:
This describes a Bernoulli distribution with parameter $p=\frac 15$, which is for an experiment with a single yes-no question.
However, we don't have a single yes-no question, but we have 10 questions.
We are looking for a distribution for the number of successes in a sequence of $n$ independent experiments, each asking a yesâ€“no question, and each with its own Boolean-valued outcome: success (with probability $p$) or failure (with probability $q = 1 âˆ’ p$). (Sweating)

So do we have a binomial distribution? :unsure:
Klaas van Aarsen said:
More or less. We can use a table with the probabilities of each possible outcome to help us answer each of the questions in (b).
But we can also find the answers for (b) in a different fashion.

Do we have the following ?

(i) $p_X(4)=P(X=4)=\binom{10}{4}p^4(1-p)^{10-4}=\binom{10}{4}\left (\frac{1}{5}\right )^4\left (\frac{4}{5}\right )^{6}$
(ii) $P(X>4)=1-P(X\leq 4)=1-\sum_{i=0}^4P(X=i)=1-\sum_{i=0}^4\binom{10}{i}p^i(1-p)^{10-i}=1-\sum_{i=0}^4\binom{10}{i}\left (\frac{1}{5}\right )^i\left (\frac{4}{5}\right )^{10-i}$
(iii) $p_X(10)=P(X=10)=\binom{10}{10}p^{10}(1-p)^{10-10}=\left (\frac{1}{5}\right )^{10}$
(iv) $P(X\geq 5)=P(X>4)=1-\sum_{i=0}^4\binom{10}{i}\left (\frac{1}{5}\right )^i\left (\frac{4}{5}\right )^{10-i}$
(v) $P(5\leq X\leq 8)=P(X\leq 8)-P(X<5)=P(X\leq 8)-P(X\leq 4)=[1-P(X> 8)]-[1-P(X> 4)]=[1-P(X=9)-P(X=10)]-\left [1-1+\sum_{i=0}^4\binom{10}{i}\left (\frac{1}{5}\right )^i\left (\frac{4}{5}\right )^{10-i}\right ]=[1-\binom{10}{9}\left (\frac{1}{5}\right )^9\left (\frac{4}{5}\right )^{10-9}-\left (\frac{1}{5}\right )^{10}]-\left [1-1+\sum_{i=0}^4\binom{10}{i}\left (\frac{1}{5}\right )^i\left (\frac{4}{5}\right )^{10-i}\right ]$

:unsure:

Yep. All correct. (Sun)

## 1. What is a random variable in a multiple choice test?

A random variable in a multiple choice test is a numerical value that represents the outcome of a randomly selected question. It can take on different values depending on the options provided in the test.

## 2. How is a random variable used in a multiple choice test?

A random variable is used in a multiple choice test to measure the performance of a test-taker. It can help determine the difficulty level of the test and provide insights on the test-taker's understanding of the subject.

## 3. What is the difference between a discrete and continuous random variable in a multiple choice test?

A discrete random variable in a multiple choice test can only take on a finite number of values, while a continuous random variable can take on any value within a given range. In a multiple choice test, a discrete random variable could represent the number of correct answers, while a continuous random variable could represent the time taken to complete the test.

## 4. How is the probability of a random variable determined in a multiple choice test?

The probability of a random variable in a multiple choice test is determined by dividing the number of times the variable occurs by the total number of possible outcomes. For example, if there are 4 options for a question and the correct answer occurs 2 times, the probability would be 2/4 or 50%.

## 5. Can a random variable be manipulated in a multiple choice test?

No, a random variable cannot be manipulated in a multiple choice test. It is determined by the options provided in the test and the test-taker's responses. However, the test creator can control the difficulty level of the test by selecting the options and their corresponding probabilities.

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