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Conditional Probability is defined as the occurrence of any event which determines the probability of happening of the other events. Let us imagine a situation, a company allows two days’ holidays in a week apart from Sunday. If Saturday is considered as a holiday, then what would be the probability of Tuesday being considered a holiday as well? To find this out, we use the term Conditional Probability.
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Key Terms: Conditional Probability,Joint Probability,Multiplication Theorem of Probability
What is Conditional Probability?
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We have discussed the definition of Conditional Probability. Let’s now understand the formula to calculate the same.
Suppose an event B has already occurred; now we have to calculate the probability of occurrence of event A. So, we can calculate the occurrence of event A on the basis of the already available information regarding the already happened event B. The conditional Probability of A would be denoted by P (A|B):
P(A/B) = P(AB)/P(B), where P(B) 0
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The video below explains this:
Conditional Probability Detailed Video Explanation:
Let’s discuss certain theorems of Conditional Probability:
- Let us consider a random experiment where the sample space S is considered as space and two events namely A and B happen there. Then, the formula would be:
P(S | B) = P(B | B) = 1.
Proof of the same: P(S | B) = P(S ∩ B) ⁄ P(B) = P(B) ⁄ P(B) = 1.
[S ∩ B indicates the outcomes common in S and B equals the outcomes in B].
- Now let us consider any two events namely A and B happening in a sample space ‘s’, then, P(A ∩ B) = P(A).
P(B | A), P(A) >0 or, P(A ∩ B) = P(B).P(A | B), P(B) > 0.
This theorem is named as the Multiplication Theorem of Probability.
Proof of the same: As we all know that P(B | A) = P(B ∩ A) / P(A), P(A) ≠ 0.
We can also say that P(B|A) = P(A ∩ B) ⁄ P(A) (as A ∩ B = B ∩ A).
So, P(A ∩ B) = P(A). P(B | A).
Similarly, P(A ∩ B) = P(B). P(A | B).
The interesting information regarding the Multiplication Theorem is that it can further be extended to more than two events and not just limited to the two events. So, one can also use this theorem to find out the conditional probability in terms of A, B, or C.
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Sometimes students get confused between Conditional Probability and Joint Probability. It is essential to know the differences between the two.
Multiplication Theorem of Probability Video Explanation
Multiplication Theorem on Probability Video Explanation
Also Read:
Joint Probability
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It is defined as the probability of occurrence of two or more events. It means it is the bisection of two or more two events. The formula to calculate the same is:
P(A ∩ B)
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Conditional Probability
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Conditional Probability on the other hand is the probability of occurrence of one event considering that the information of the already happening event is available. So, here we have the information regarding one event let's say A which has already happened, and using that available information we will find the probability of happening of event B. But in Joint Probability, we don’t have any already available information!
Students are advised to understand the basic concept about Conditional Probability and certain theorems related to the same. After that, it would be easier to solve the questions on the same topic.
Let us now discuss some frequently asked questions on Conditional Probability in the last section of the article.
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Sample Questions based on Conditional Probability
Ques. Does it mean that both the events happen at the same time while calculating the Conditional Probability? (2 marks)
Answer: No, both the events don’t happen at the same time while calculating the conditional probability. One event has already taken place and we are using this information to find the occurrence of another event that will happen later on.
Ques. Is there any real-life example that exhibits Conditional Probability?(2 marks)
Answer: Yes, there is a practical application of Conditional Probability in daily life. Let’s consider that as per the Weather Forecast, the probability of rainfall is 50%. But this depends on the information available on various factors like wind intensity, temperature, and quantitative analysis of humidity. Thus, Conditional Probability can be used in real life!
Ques. Mention any two properties of Conditional Probability.(2 marks)
Answer: Properties of Conditional Probability are given below:
- Let us imagine that A and B are the events of sample space named S, then the formula would be
P(A|B) = P(B|B) = 1.
- Another property can be represented by the formula given below:
P(E′|F) = 1 − P(E|F), where E and F are the two events
Ques. A manufacturer has three machine operators A, B, and C. The first operator A produces 1% of defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job 30% of the time and C on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random .from his and is found to be defective. What is the probability that it was produces by A? (2019)
Solution:
Let’s take H? as the event items produced by A
H? as B’s event item produced
H? as C’s event items produced
Ques. If P(not A) = 0·7, P(B) = 0·7 and P(B/A) = 0·5, then find P(A?B). (2019 outside Delhi)
Solution:
Ques. There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is the two-headed coin? (2019 outside Delhi)
Solution:
Given, three coins
So let’s take,
E? as the two-headed coin
E? as the biased coin
E? as the unbiased coin
A as the which shows only the head
Ques. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’, is obtained. If she obtained exactly one “tail7, what is the probability that she threw 3, 4, 5 or 6 with the dice? (2018)
Solution:
The sample space shows 36 outcomes. Let’s take A as the event that the sum total of the observations 8.
Therefore A = {(2,6)(3,5)(5,3)(4,4)(6,2)}
Ques. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. (2018)
Solution:
Let’s take E2 as the event that the girl obtains 1 or 2 on the dice roll and E2 as the event that the girl obtains 3, 4, 5, or 6 on the dice role:
Therefore P(E2 ) = 26= 13
P(E?) = 46= 23
Let’s take A as the event that she obtains exactly one tail. If she tossed the coin thrice and obtains exactly one tail then possible outcomes are HTH, HHT, THH
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