Multiple choice test : random variable

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
  • #1
mathmari
Gold Member
MHB
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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,
(iii) all tasks 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:
 
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  • #2
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. 🤔
 
  • #3
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:
 
  • #4
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.
 
  • #5
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:
 
  • #6
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 yes–no question, and each with its own Boolean-valued outcome: success (with probability $p$) or failure (with probability $q = 1 − p$). (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. 🤔
 
  • #7
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 yes–no question, and each with its own Boolean-valued outcome: success (with probability $p$) or failure (with probability $q = 1 − p$). (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:
 
  • #8
Yep. All correct. (Sun)
 

Related to Multiple choice test : random variable

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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