Probability

Test your knowledge of permutations and combinations with this interactive quiz! The MCQs Permutations Combinations Quiz covers essential concepts like factorials, combinations, arrangements, and real-world applications. This quiz is perfect for students and enthusiasts looking to sharpen their probability and counting skills. Let us start with the MCQs Permutation Combinations Quiz now.

Learn about Counting Techniques in Probability

Online MCQs Permutations Combinations

• The number of ways to select 2 persons from 6, ignoring the order of selection, is
• $n!=$?
• An arrangement of all or some of a set of objects in a definite order is called
• An arrangement of objects without caring for the order is called
• ${}^nP_r$ =
• ${}^nC_r$ =
• In how many ways can a team of 6 players be chosen from 11 persons
• How many terms are in the expansion of the $(q+p)^n$
• ${}^{10}C_5=$
• ${}^5C_5$ is equal to
• The difference between permutation and combination lies in the fact that
• Which of the following statements is true?
• A homeowner doing some remodeling requires the services of both a plumbing contractor and an electrical contractor. If there are 15 plumbing contractors and 10 electrical contractors available in the area, in how many ways can the contractors be chosen?
• How many permutations of size 3 can be constructed from the set (A, B, C, D, E)?
• How many combinations of size 4 can be formed from a set of 6 distinct objects?
• An experiment consists of three stages. There are five ways to accomplish the first stage, four ways to accomplish the second stage, and three ways to accomplish the third stage. The total number of ways to accomplish the experiment is
• The $0!$ is
• In how many ways can be letters in the word UNIVERSITY be arranged randomly
• Seventeen teams can take part in the Football Championship of a country. In how many ways can the Gold, Silver, and Bronze medals be distributed among the teams?
• The number of 3-digit telephone area codes that can be made if repetitions are not allowed is

See Recursion in R Language

Probability is a fundamental concept in statistics used to quantify uncertainty. One of the key concepts in probability is the Complement of an event. The complement of an event provides a different perspective on computing the probabilities, that is, it is used to determine the likelihood of an event not occurring. Let us explore how the complement of an event is used for the computation of probability.

What is the Complement of an Event?

The complement of an event $E$ is denoted by $E’$, encompasses all outcomes in the sample space that are not part of event $E$. In simple terms, if event $E$ represents a specific outcome or set of outcomes, its complement represents everything else that could occur.

For example, let the event $E$ be rolling a 4 on a six-sided die; the complement of event $E$ is ($E’$) rolling a 1, 2, 3, 5, or 6.

Note that event $E$ and its complement $E’$ cover the entire sample space of the die roll.

Complement Rule: Calculating Probabilities

A pivotal property of complementary events is that the sum of their probabilities is 1 (or 100%). This is because either the event happens or it does not happen, as there are no other probabilities. It can be described as
$$P(E) + P(E’) = 1$$
This leads to the complement rule, which states that
$$P(E’)= 1- P(E)$$
It is useful when computing the probability of an event not occurring.

Examples (Finding the Complement of an Event)

Suppose the probability that today is a rainy day is 0.3. The probability of it not raining today is $$1-0.3 = 0.7$$

Similarly, the probability of rolling a 2 on a fair die is $P(E) = \frac{1}{6}$. the probability of not rolling a 2 is $P(E’)=1-\frac{1}{6} = \frac{5}{6}$.

Why use the Complement Rule?

Sometimes, calculating the probability of the complement is easier than calculating the probability of the event itself. For example,

Question: What is the probability of getting at least one head in three coin tosses?
Solution: Instead of listing all possible favourable outcomes, one can easily use the complement rule. That is,
Complement Event: Getting no heads (all tails)
Probability of all tails = $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$. Therefore, the probability of at least one head is

P(At least one head) = $1 – \frac{1}{8} = \frac{7}{8}$
This approach is quicker than counting all possible cases; that is, one can avoid enumerating all the favourable outcomes.

Properties of Complementary Events

• Mutually Exclusive: An event and its complement cannot occur together (simultaneously)
• Collectively Exhaustive: An event and its complement encompass all possible outcomes
• Probability Sum: The probabilities of an event and its complement add up to 1.

Understanding complements in probability can make complex problems much simpler and easier.

Practical Applications

Understanding complements is invaluable in various fields:

• Quality Control: Determining the probability of defects in manufacturing
• Everyday Decisions: Estimating probabilities in daily life, such as the chance of missing a bus or the likelihood of rain.
• Game Theory: Calculating chances of winning or losing scenarios
• Risk Assessment: Evaluating the likelihood of adverse events not occurring

More Examples (Complement of an Event)

• In a standard 52-card deck, what is the probability of not drawing a heart card?
  $P(Not\,\,Heart) = 1 – P(Heart) = 1 – \frac{13}{52} = \frac{39}{52}$
• If the probability of passing an examination is 0.85, what is the probability of failing it?
  $P(Fail) = 1 – P(Pass) = 1 – 0.85 = 0.15$
• If the probability that a flight will be delayed is 0.13, then the probability that it will not be delayed will be $1 – 0.13 = 0.87$
• If $k$ is the event of drawing a king card from a well-shuffled 52-card deck, then the event $K’$ is the event that a king is not drawn, so $K’$ will contain 48 possible outcomes.

Data Analysis in the R Programming Language

The post is about the Online MCQs Probability Quiz Questions. There are 20 multiple-choice questions covering topics related to random experiments, random variables, probability, classical and empirical probability, rules of probability (addition and multiplication rule), conditional probability, events, types of events, and sample space. Let us start with the MCQs Probability Quiz Questions now.

Please go to MCQs Probability Quiz Questions 13 to view the test

Online MCQs Probability Quiz Questions with Answers

• Rolling a 6-faced die, what’s the probability of seeing a “6”?
• Rolling a 6-faced die, given that the number is even, what’s the probability that we’ve got a “6”?
• Rolling a 6-faced die, what’s the probability of seeing an even number?
• Rolling two independent 6-faced dice, what’s the probability that both dice show the same number?
• Which of the following corresponds to the formula for the discrete form of Bayes’ Rule?
• The license plates in a certain country bear six characters taken at random among the 26 letters of the alphabet and the ten digits ${0, 1,\cdots,9}$. What is the probability that a given license plate bears at least one digit?
• Let $A$, $B$, and $C$ be events such that $A \subset B$, $A$, and $C$ are incompatible, $P[(B \cup C)^c] = \frac{1}{10}$, $P[B\cap C] = \frac{3}{10}$, $P[A’\cap B] = \frac{1}{2}$ and $P[C] = \frac{13}{20}$. Calculate $P[A]$.
• The events $A$ and $B$ form a partition of the sample space $S$. Calculate $P[A|B] + P[A|B^c]$.
• If a card is chosen from a standard deck of cards, what is the probability of getting a diamond or a club?
• If $P(B|A)=0.25$  and $P(A\cap B)=0.2$ then $P(A)$ will be ———–.
• A coin is so weighted that $P(T)=2P(H)$ the $P(H)$ is ———–.
• A set of events is said to be independent if the probability of head on tossing a coin is ————-?
• Two events are said to be equally likely is ————-?
• A subset of the sample space is called ————?
• The collection of one or more outcomes from an experiment is called ————–?
• —————– are said to be exhaustive if they constitute the entire sample space?
• Events occurring together without affecting each other are called ————-?
• If the events have same chance of occurrence, the events are called —————?
• If in a coin $P(H)=2P(T)$ then the probability of a head is ————-?
• A box containing 12 balls of color, 6 red and 6 white. a white ball is selected. What is the probability of white ball —————?

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