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Basic Probability Theory Trivia Quiz

This quiz tests your knowledge on some basic probability theory concepts. Have fun and thanks for playing.

A multiple-choice quiz by Matthew_07. Estimated time: 5 mins.

Author
Matthew_07

Time
5 mins

Type
Multiple Choice

Quiz #
293,008

Updated
Dec 03 21

# Qns
10

Difficulty
Tough

Avg Score
5 / 10

Plays
873

One at a Time - Single Page - Timed Game

Question 1 of 10

1. In combinatorial analysis, the basic principle of counting provides a convenient way to calculate the number of possible outcomes for experiments. Let say you are given 3 caps, 4 shirts, 5 pants and 6 pairs of shoes. By using this principle, how many ways can you dress yourself? Hint

3 x 4 x 5 x 6 = 360

3! + 4! + 5! + 6! = 870

3 + 4 + 5 + 6 = 18

3! x 4! x 5! x 6! = 12441600

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Question 2 of 10

2. Let say there are 8 contestants in a contest. There are 8P3 = 336 possible combinations for the top three spots. Here, the letter "P" stands for permutation. Which of the following formula is equivalent to nPr? Hint

n x (n - 1) x ... x (n - r)

n x (n - 1) x ... x (n - r - 1)

n x (n - 1) x ... x 3 x 2 x 1

n x (n - 1) x ... x (n - r + 1)

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Question 3 of 10

3. You are given 10 balls and you are to choose 3 balls from these 10 balls. So, you have 10C3 = 120 ways to choose it. Here, the letter "C" represents "combination". Is it true that nCr = nC(n-r)?

Yes

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Question 4 of 10

4. The binomial theorem provides a convenient way to calculate the value of the coefficients for all the terms in any expansion involving 2 unknowns. The values of these coefficients can also be obtained from which famous mathematical figures? Hint

Pentagram

Pascal's triangles

Pythagorean triplets

Permutation tree

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Question 5 of 10

5. Set operations are used extensively in the study of probability theory. Let A, B and C be any 3 events and S is the sample space. Choose the WRONG matching pair from the list below. Hint

Distributive Law: A union (B intersect C) = (A union B) intersect (A union C)

Identity Law: A union S = S

Associative Law: A union B = B union A

De Morgan's Law: (A union B)' = A' intersect B'

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Question 6 of 10

6. In general, there are 4 definitions for probability. Which one of these 4 definitions is the one that is used as the fundamental and formal definition in the study of probability theory? Hint

A measure of belief

Relative frequency or posteriori probability

Classical or priori probability

The axiomatic definition of probability

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Question 7 of 10

7. A conditional probability is represented by P(A|B). How do we interpret this probability? Hint

The probability of the event B happens given that the event A has happened.

The probability of the event A happens given that the event B has happened.

Both of these.

None of these.

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Question 8 of 10

8. Let say 2 events satisfy the following equation: P(A intersect B) = P(A) x P(B). We say that events A and B are ___. Hint

Disjoint

Dependent

Independent

Mutually exclusive

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Question 9 of 10

9. If events A and B are independent, will events A' and B' be independent as well?

Yes

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Question 10 of 10

10. There are 3 boxes, A, B and C. The probability of choosing box A, B and C are 0.5, 0.3 and 0.2 respectively. Box A, B and C contains 20%, 30% and 50% rotten apples. What is the probability that an apple is drawn from box A given that it is rotten? To solve this kind of problem, what theorem should we use? Hint

Beck's Theorem

Bayes' Theorem

Binomial Theorem

Bonnet's Theorem

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Quiz Answer Key and Fun Facts

Answer: 3 x 4 x 5 x 6 = 360

The basic principle of counting uses only the multiplication operation. The factorial operation is used in combinations and permutations.

Answer: n x (n - 1) x ... x (n - r + 1)

On the other hand, the formula for the factorial operation is given by n! = n x (n - 1) x ... x 3 x 2 x 1.

Answer: Yes

Notice that nCr = n!/[(n-r)!r!]. Also, nC(n-r)= n!/[(n-r)!(n-(n-r))!] = n!/[(n-r)!r!]= nCr. Therefore, nCr and nC(n-r) are equivalent.

Answer: Pascal's triangles

We can obtain the values of the coefficients quickly by using the numbers generated from the Pascal's triangle. For example, the third row has the numbers 1, 2, 1, which corresponds to the answer (x + y)^2 = x^2 + 2xy + y^2. A more general theorem is the multinomial theorem, which can be applied for the expansions that involve 2 or more unknowns.

5. Set operations are used extensively in the study of probability theory. Let A, B and C be any 3 events and S is the sample space. Choose the WRONG matching pair from the list below.

Answer: Associative Law: A union B = B union A

Actually, A union B = B union A refers to the commutative law. The associative law is (A union B) union C = A union (B union C).

6. In general, there are 4 definitions for probability. Which one of these 4 definitions is the one that is used as the fundamental and formal definition in the study of probability theory?

Answer: The axiomatic definition of probability

In mathematics, axioms are definitions defined by mathematicians. We do not prove axioms because they are simply "the rules of the game", but we use axioms to prove theorems.

7. A conditional probability is represented by P(A|B). How do we interpret this probability?

Answer: The probability of the event A happens given that the event B has happened.

The formula is given by P(A|B) = P(A intersect B) / P(B), and P(B) must be greater than 0.

8. Let say 2 events satisfy the following equation: P(A intersect B) = P(A) x P(B). We say that events A and B are ___.

Answer: Independent

By definition, if P(A intersect B) = P(A) x P(B), then those 2 events are independent of each other. Otherwise, they are dependent.

Mutually exclusive events has the same meaning as disjoint events. Its characteristics is P(A intersect B) = 0. In other words, events A and B share no common point or area in the Venn diagram.

9. If events A and B are independent, will events A' and B' be independent as well?

Answer: Yes

Events A and B are independent implies that these 2 events satisfy the equation P(A intersect B) = P(A) x P(B). It can be proved that P(A' intersect B') = P(A') x P(B'). Starting from P(A' intersect B'), we have P(A' intersect B') = 1 - P(A union B) = 1 - [P(A) + P(B) - P(A intersect B)] = 1 - P(A) - P(B) + P(A) x P(B) = 1 - P(A) - P(B) [1 - P(A)] = [1-P(A)][1-P(B)] = P(A') x P(B').

Answer: Bayes' Theorem

Let A = event that an apple is drawn from box A, B = event that an apple is drawn from box B, C = event that an apple is drawn from box C, R = a rotten apple is drawn.

Given P(A) = 0.5, P(B) = 0.3, P(C) = 0.2.
Also, P(R|A) = 0.2, P(R|B) = 0.3, P(R|C) = 0.5.
Our aim is to find P(A|R) = P(A intersect R)/P(R)
= P(A) x P(R|A) / [P(A) x P(R|A) + P(B) x P(R|B) + P(C) x P(R|C)]
= 0.5 x 0.2 / (0.5 x 0.2 + 0.3 x 0.3 + 0.2 x 0.5)
= 0.3448.

Bayesian inference is a branch of statistic inference that arises from this theorem.

Source: Author Matthew_07

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