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
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 —————?
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.
Table of Contents
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
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
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.
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%.
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%.
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%.
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%.
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%.
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%.
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%.
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%.
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%.
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
Write down the conditional probability formula.
Describe the numerator and denominator in the conditional probability formula.
Give some real-life examples that make use of the conditional probability formula.
What is a prior probability.
How conditional probability can be used to make informed decisions and predictions? Explain.
In this post, I will discuss the Basics of Probability theory. First I will start with the concept of Set and Event.
Table of Contents
Set
In statistical theory, a set is a well-defined collection of distinct events. For example, whenever a coin is tossed or die rolled, something (event) will happen. Distinct events comprise the set, that is when a coin is tossed, either Hear or Tail. It can be denoted with a Set.
$$A=\{Head, \, Tail\}$$
Similarly, for a fair die, the distinct events can be represented as set $B$, that is,
$$B = \{1, 2, 3, 4, 5, 6\}$$
When two fair dice are rolled, there will be 36 events that can be represented in a set say $C$.
Probability is the chance of occurrence of an event described in a set (or sample space). For example, what is the chance of rain today? what is the chance that Pakistan will win the T20 World Cup? Probability is the estimation of chance and it deals with the occurrence of an event in the future. The estimates are presented numerically. For example, (i) There is a 75% chance of rain today, (ii) The insurance industry requires precise knowledge about the risk of loss to calculate premiums, and (iii) The chances of winning the lottery game are 1 in 2.3 million.
Random Experiment
Regarding probability, it is important to understand the concept of random experiments. It is a planned process/activity that gives different results known as outcomes. For example, as discussed earlier, when a coin is tossed, there may be two possible outcomes, Head or Tail. Any experiment or planned process which has only one outcome cannot be regarded as a random experiment. A random experiment has at least a minimum of two outcomes. Outcomes are the results of the experiment. The random experiment has the following properties:
It can be repeated any number of times practically or theoretically.
Each experiment has a minimum of two possible outcomes.
All the outcomes are known in advance but each outcome is unpredictable.
So, we can say that probability is the measure of the degree of uncertainty or quantification of uncertainty.
Sample Space
When we collect all possible outcomes, it is known as sample space, and represented by $S$. For example,
$S=\{H, T\}$ sample space for tossing a single coin
$S=\{HH, HT, TH, TT\}$ sample space when tossing two coins simultaneously
Each outcome of a sample space is called a sample point.
Event
The individual outcome from a sample space in which one is interested is called an event. Events may be based on a single sample point or more than one sample point. For example, Let the even be even numbers when a single dice is thrown, that is, $A=\{2, 4, 6\}$, or even maybe $T$ (Tail) when tossing a coin. $B=\{T\}$. When we throw two dice the event may be the same number on the upper face of the dice, $C=\{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\}$
Similarly, the sum of dots on the top face of two dice is 4 is another event, that is, $D=\{(2, 2), (1, 3), (3, 1)\}$
Types of Events
The probability of an event lies between 0 and 1 inclusive. If the probability of an event is 1, it is known as a sure event. If the probability of an event is zero it is an impossible event. When two or more events cannot occur at the same time it is called a mutually exclusive event. For example, in the coin tossing example, either $H$ will occur or Tail, both head and tail cannot occur at the same time.
Events are equally likely when events have the same chance of occurrence. For example, either a student will pass or fail, there is a 50% chance for both events. Collectively Exhaustive Events are events whose union is equal to the sample space.
Random Variable
A random variable is that which takes values randomly. A random variable may be represented by $X$, $Y$, and $Z$, etc. Random variables can be classified as discrete random variables or continuous random variables. A discrete random variable is based on a counting procedure, while a continuous random variable is based on measurements.
A random variable is a variable that takes values randomly. These values may be integers for discrete variables and real for continuous variables. When we toss a coin there may be $H$ or $T$. Suppose, you are interested in Head then the random variable may be denoted by $X$ for various numbers of heads (for example, 0 head and 1 head)
