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Conditional probability formula is most closely related to Bayes' theorem, one of statistics' most influential theories. The likelihood of one event occurring in conjunction with one or more other events is referred to as conditional probability. The conditional probability formula calculates the likelihood of an event, say B, occurring given the occurrence of another event, say A. In this article, we will look into the derivation of the conditional probability formula along with suitable examples.
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Key Terms: Probability, conditional probability, formula of conditional probability, prediction of outcomes, probability theory
Also Read: Linear Programming
Conditional Probability Formula
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One of the most fundamental notions in probability theory is the conditional probability formula. The conditional probability formula calculates the likelihood of an event, say B, occurring given the occurrence of another event, say A.
Conditional Probability
The video below explains this:
Conditional Probability Detailed Video Explanation:
Event A: Today's chance of rain is 0.4 percent (40 percent).
Event B: I'm going outside, with a 0.5 chance of happening (50 percent).
A conditional probability considers the likelihood of both rain and going outside in relation to one another. Let's look at a few examples that explain the conditional probability formula. Please keep in mind that conditional probability does not always imply a causal relationship between the two events, nor does it imply that they occur at the same time.
By knowing the conditional probability of event B, given that event A has occurred, as well as the individual probabilities of events A and B, the Bayes' theorem is used to predict the conditional probability of event A, given that event B has occurred.
Conditional Probability Formula
The conditional probability of P (A | B) is undefined when P(B)=0. (Event B did not take place)
The conditional probability formula is:
P (A | B) = P (A and B) / P(B)
It's also possible to write it as,
P(A|B) = P(A∩B) P(B)
Also Read:
Derivation of Conditional Probability Formula
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The probability multiplication rule is used to create the conditional probability formula.
P(A) → Probability of Event A Occurring
P(B) → Probability of Event B Occurring
P(AB) → The occurrence of both events A and B, or the common elements of both events.
Event A already has taken place.
If B has also occurred, those outcomes that are not in B but are in A are deleted, narrowing the sample space to set B.
The possible outcomes for A and B are thus limited to those in which B occurs, so A can only happen if the outcome belongs to the set A ∩ B.
As a result, we divide P (A ∩ B) by P(B), effectively limiting the sample space to circumstances in which B occurs.
Thus we arrive at the formula for conditional Probability as,
P(A|B) = P(A∩B) P(B)
Application of Conditional Probability Formula
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- The prediction of outcomes in the cases of flipping a coin, choosing a card from a deck, and throwing dice are only a few of the most common applications of the conditional probability formula.
Prediction Outcomes
- It also aids Data Scientists in obtaining better outcomes when analysing data sets.
- It assists machine learning engineers in creating more accurate prediction models.
Also Read:
Things to Remember
- The likelihood of one event occurring in conjunction with one or more other events is referred to as conditional probability
- The conditional probability formula calculates the likelihood of an event, say B, occurring given the occurrence of another event, say A.
- The conditional probability of P (A | B) is undefined when P(B)=0.
- The conditional probability formula is P (A | B) = P (A and B)/P(B).
Previous Year Questions
- A number xx is chosen at random from the set {1,2,3,4,.....,100}[JEE Main 2014]
- If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :[JEE Main 2018]
- Then the expected value of X, is :..[JEE Main 2020]
- If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/ loss (in rupees) is :….[JEE Main 2019]
- If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is : [JEE Main 2019]
- If two different numbers are taken from the set {0,1,2,3,……,10}then the probability that their sum as well as absolute difference are both multiple of 4, is :..[JEE Main 2017]
- Then the probability that one of the boxes contains excatly 3 balls, is...[JEE Main 2015]
- The probability that the card was drawn from Box I is :...[JEE Main 2020]
- Then , the events A and B are….[JEE Main 2014]
- The probability that one person speaks Hindi only and the other speaks both Hindi and English is….[KEAM]
- The chance of getting a total of 12 in 5 throws is...[KEAM]
- At a randomly chosen time, the probability that the light will not be green, is….[KEAM]
- If A and B are mutually exclusive events and if p(B)=13,p(A∪B)=1321, then P(A) is equal to….[KEAM]
- The probability that the second ball is red, is :...[JEE Main 2019]
- If X be the number of white balls drawn, the ((meanofXstandarddeviationofX) is equal to :...[JEE Main 2019]
- If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :...[JEE Main 2017]
- If the mean and the variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than or equal to one is :...[JEE Main 2015]
- An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :….[JEE Main 2017]
- If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :...[JEE Main 2019]
- The probability of at least 55 successes in the six trials of this experiment is :….[JEE Main 2016]
Sample Questions
Ques. Four people bought apples, three people bought oranges, and two people bought apples and oranges in a group of ten. What is the probability that a consumer bought apples and oranges at the same time using the conditional probability formula? (3 marks)
Ans. People who bought apples should be labelled A, and those who bought oranges should be labelled O.
Given that,
P(A) = four out of ten = forty percent = 0.4
P(O) = 3 out of 10 = 30% = 0.3
Hence,
P(A∩O) = 2 out of 10 = 20% = 0.2
Now, apply the conditional probability formula to the situation.
P(O|A) = P(A∩O) / P(A) = 0.2 / 0.4 = 0.5 = 50 percent P(O|A) = P(AO) / P(A) = 0.2 / 0.4 = 0.5 = 50 percent
Given that a consumer purchased apples, there is a 50% chance that they purchased oranges.
Ques. My next-door neighbour has two children. I find that she has a son named Adam. What are the chances that Adam's younger brother or sister is a boy? (3 marks)
Ans. Let B be the boy child and G be the girl child.
S = BB, BG, GB, GG is the sample space.
Assume that males and girls have equal chances of being born and that the four elements of S are also equal chances.
The set X = BB, BG, GB represents the event, X, that the neighbour has a son.
As a result, P(X) = 3/4.
The set Y = BB represents the event, Y, that the neighbour has two sons.
P (Y∩ X) = 1/4 then.
Now, apply the conditional probability formula to the situation.
P (Y | X) = P (Y∩ X) / P(X) = (1/4) / (3/4) = 1/3
Ques. The dice are rolled fairly. What is the likelihood of A given B, where A is the probability of getting an even number and B is the probability of having a number less than or equal to 2? (3 marks)
Ans. Using the information provided, find P (A | B).
The sample space = 1, 2, 3, 4, 5, 6 when a die is rolled.
The event of getting an even number is referred to as A. As a result, A = 2, 4, 6.
Event B is when you get a number that is less than or equal to 3. As a result, B = 1, 2.
As a result, A∩ B = 2.
Using the conditional probability formula, we can now:
P (A | B) = P (A ∩ B) / P (B)
P (A | B) = (1/6)/ (3/6) = 1/3
A given B has a 1/3 chance of happening.
Ques. There is a 0.03 chance that it is Friday, and a pupil is absent. Because there are five school days in a week, the chances of it being Friday are 0.2. Given that it is Friday, what is the likelihood that a student will be absent? (2 marks)
Ans. Conditional probability's formula is as follows:
P(B/A) = P (A ∩ B)/P(A)
P (Absent | Friday) = P (Absent and Friday)/P(Friday)
= 0.03/0.2
equals 0.15
15 percent
Ques. A teacher handed her pupils two tests, one in arithmetic and one in science. The maths test was passed by 40% of the pupils, while 25% of the students passed both tests. How many people who passed the math exam also passed the science exam? (3 marks)
Ans. Given,
The percentage of students that passed the math exam was 40%.
The percentage of pupils who passed both assessments was 25%.
Let A and B be the number of students who passed math and science exams, respectively.
According to the information provided,
P(A) = 0.40
P (A∩B) = 0.25
Students who passed the arithmetic test also passed the science test in a high percentage of cases.
= B's conditional probability given A's conditional probability
= P(B|A)
= P (A ∩ B)/P(A)
= 0.25/0.40
= 0.625
= 62.5%
Ques. Green and yellow balls are contained in a bag. A total of two balls are chosen without being replaced. The odds of picking a green ball first and subsequently a yellow ball are 0.28. On the first draw, there is a 0.5 chance of selecting a green ball. Given that the first ball drawn was green, calculate the chance of selecting a yellow ball on the second draw. (3 marks)
Ans. Let A and B represent the first and second draws, respectively, where a green ball was drawn and a yellow ball was drawn.
Using the information provided,
P(A) = 0.5
P(A ∩ B) = 0.28
Given that the first ball drawn was green, the probability of choosing a yellow ball on the second draw = Conditional of B given A
=P(B|A)
= P (A∩ B)/P(A)
= 0.28/0.5
equals 0.56
Ques. What is the proof for the conditional probability formula, P(A/B)=P(A*B) /P(B), that is, P(A/B)=P(A*B) /P(B)? (3 marks)
Ans. Allow an n-times repetition of a random experiment.
Let event A occur n1 times, event B n2 times, and event A*B n3 times in these n repetitions.
The phrase n3/n2 denotes the relative frequency of A among individuals who have experienced B.
It indicates that n3/n2 (assuming n2>0) is the conditional relative frequency of A when B occurs.
Then (n3/n)/(n2/n) can be represented as n3/n2.
The probability P(A*B) is the numerator of this formula in the limit as n approaches infinity, and the denominator, P, is the denominator under the same conditions (B).
As a result, P(A*B)/P(B) becomes the definition of conditional probability P(A/B), which is the likelihood of A occurring if B has already happened.
Ques. What is the Conditional Probability Formula Used For? (2 marks)
Ans. Prediction of outcomes is one of the applications of the conditional probability formula.
- Tossing a coin
- A card from the deck is chosen
- Throwing dice
- Data Scientists use this when analysing a set of data.
- Machine Learning Engineers use it to create more precise prediction models.
Ques. Black and white marbles are kept in a jar. Two marbles are picked at random and will not be replaced. Selecting a black marble and then a white marble has a probability of 0.34 and selecting a black marble on the first draw has a probability of 0.47. Given that the first marble selected was black, what is the likelihood of selecting a white marble on the second draw? (2 marks)
Ans. Given,
P (Black and white) =0.34
P (Black)= 0.47
We know that,
P (White | Black) = P (Black and White)/ P(Black)
= 0.34/0.47
= 0.72
= 72%
Ques. A student at Roosevelt Middle School has a 0.087 chance of taking Technology and Spanish. A student's chance of taking Technology is 0.68. What are the chances that a student will take Spanish if they are also taking Technology? (2 marks)
Ans. Given,
P (Technology and Spanish) = 0.087
P (Technology) = 0.68
We know that,
P (Spanish | Technology) = P (Technology and Spanish)/ P (Technology)
= 0.087/0.68
= 0.13
= 13%
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