Probability

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.

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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.

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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In this pose, we will discuss the conditional probability formula and related real-life practical examples. Conditional probability is a fundamental concept in probability theory that deals with the likelihood of an event occurring given that another event has already happened.

Introduction Conditional Probability

A conditional probability for an event is computed when some additional (prior) information about the experiment’s outcome is known. Suppose, there are two events $A$ and $B$ for an experiment. Also, suppose that it is known that event $B$ has occurred. One can calculate the probability of Event $A$ based on the formation of the event $B$. This probability is called the conditional probability of $A$ given $B$. The conditional probability of $A$ given $B$ is denoted by $P(A|B)$.

The Conditional Probability Formula

For event $A$ and $B$, where the event $B$ has already occurred, the Conditional Probability Formula can be described as

$$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad provided\,\, that\,\, P(B)\ne 0 $$

• $P(B∣A)$ is the probability of event $B$ occurring given that event $A$ has occurred.
• $P(A∩B)$ is the probability of both events $A$ and $B$ occurring.
• $P(A)$ is the probability of event $A$ occurring.

Numerical Example of Conditional Probability

Consider the experiment of drawing two cards from a standard deck of cards without replacement. Let event $A$ be the event that the first card drawn is kind, and we are interested in calculating the probability of the event $B$ the second card drawn will also be a king card provided that the first card drawn was a king. We can conclude here that

\begin{align*}
P(A) &= P(\text{first card drawn is king}) =\frac{\text{Total number of cards}}{\text{Number of kings}}​=\frac{4}{52}=\frac{1}{13}\\
P(A∩B) &=\frac{4}{52}\times \frac{3}{51} ​= \frac{1}{221}\\
P(B|A) &= \frac{P(A\cap B}{P(B)} = \frac{1/221}{1/13} = \frac{1}{17}​
\end{align*}

Note that in the conditional probability Formula, $P(A∩B)$ is the probability that the first card is a king and the second card is also a king. Since the first card drawn is a king, there are now 3 kings left in the remaining 51 cards.

Therefore, the probability that the second card drawn is a king, given that the first card drawn was a king, is $\frac{1}{17}$.

Real-Life Examples of Conditional Probability

The following are some important real-life and practical examples of conditional probability from various fields of life.

1. Election Polling: A pollster predicts the outcome of an election. The probability that a voter will support a candidate given that they belong to a specific demographic group (e.g., age, gender, income level). For example, If 60% of voters aged 18-24 support Candidate A, then the conditional probability of supporting Candidate A given that the voter is aged 18-24 is 60%.
2. Weather Forecasting: A weather forecast is used to predict the probability of rain. One can compute the probability that it will rain given that the sky is cloudy using the conditional probability formula. For example, suppose that the historical data shows that it rains 30% of the time when the sky is cloudy, then the conditional probability of rain given cloudy skies is 30%.
3. Sports Analytics: A basketball player takes a shot. The likelihood of an event that the player makes the shot given that they are shooting from a specific distance. For example, if a certain player makes 40% of their three-point shots, then the conditional probability of making a shot given that it is a three-point attempt is 40%.
4. Customer Behavior: A retail store analyzes customer purchasing behavior. They can find the probability that a customer will buy a product given that they have viewed it online. As an example, suppose that 10% of customers who view a product online end up purchasing it, then the conditional probability of a purchase given that the product was viewed online is 10%.
5. Quality Control in Manufacturing: Items produced in a factory can be either defective or non-defective. The likelihood of an event that an item produced is defective given that it was produced by a specific machine will make use of the conditional probability formula. For example, if Machine A produces defective items 5% of the time, then the conditional probability that an item is defective given that it was produced by Machine A is 5%.
6. Traffic Light Timing: A city adjusts the timing of traffic lights to reduce congestion. The conditional probability can be used to compute the probability that a car will stop at a red light given that it is during rush hour. For example, if 70% of cars stop at a red light during rush hour, then the conditional probability of stopping at a red light given that it is rush hour is 70%.
7. Spam Filtering: An email service filters out spam emails. The conditional probability formula can be used to compute the probability that an email is spam given that it contains certain keywords (e.g., “free,” “win,” “prize”). For example, if 90% of emails containing the word “free” are spam, then the conditional probability that an email is spam given that it contains the word “free” is 90%.
8. Insurance Risk Assessment: Insurance companies assess the risk of insuring a driver. One can find the probability that a driver will have an accident given that they are under 25 years old. For example, If statistics show that drivers under 25 are involved in 20% of all accidents, then the conditional probability of an accident given that the driver is under 25 is 20%.
9. Credit Scoring: The bank assesses the creditworthiness of a loan applicant. The conditional probability is used to compute the probability that an applicant will default on a loan given that they have a low credit score. As an example, suppose, 15% of applicants with a credit score below 600 default on their loans, then the conditional probability of default given a low credit score is 15%.
10. Medical Testing: Suppose, A patient takes a medical test to determine if they have a certain disease. The probability that the patient has the disease given that the test result is positive. As an example, consider the prevalence of a disease is 1% in the population, and the test has a 99% accuracy rate (both true positive and true negative rates are 99%). The conditional probability that a person has the disease given a positive test result can be computer using the Bayes’ Theorem.

These real-life examples of conditional probability illustrate how conditional probability is used to make informed decisions and predictions in various fields of life by considering the relationship between different events or conditions.

FAQs about Conditional Probability and Conditional Probability Formula

1. Write down the conditional probability formula.
2. Describe the numerator and denominator in the conditional probability formula.
3. Give some real-life examples that make use of the conditional probability formula.
4. What is a prior probability.
5. How conditional probability can be used to make informed decisions and predictions? Explain.

Probability in R Language