Conditional Probability
Probability is a branch of Mathematics which deals with the study of occurrence of an event. There are several approaches to understand the concept of probability which include empirical, classical and theoretical approaches. The conditional probability of an event is when the probability of one event depends on the probability of occurrence of the other event. When two events are mutually dependent or when an event is dependent on another independent event, the concept of conditional probability comes into existence.
Conditional Probability Definition:
Conditional probability of occurrence of two events A and B is defined as the probability of occurrence of event ‘A’ when event B has already occurred and event B is in relation with event A.
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The above picture gives a clear understanding of conditional probability. In this picture, ‘S’ is the sample space. The circles A and B are events A and B respectively. The sample space S is restricted to the region enclosed by B when event B has already occurred. So, the probability of occurrence of event A lies within the region of B. This probability of occurrence of event A when event be has already existed lies within the region common to both the circles A and B. So, it can be denoted as the region of A ∩ B.
Conditional Probability Examples:
The man travelling in a bus reaches his destination on time if there is no traffic. The probability of the man reaching on time depends on the traffic jam. Hence, it is a conditional probability.
Pawan goes to a cafeteria. He would prefer to order tea. However, he would be fine with a cup of coffee if the tea is not being served. So, the probability that he would order a cup of coffee depends on whether tea is available in the cafeteria or not. So, it is a conditional probability.
It will rain at the end of the hottest day. Here, the probability of occurrence of rainfall is depending on the temperature throughout the day. So, it is a conditional probability.
In a practical record book, the diagrams are written with a pencil and the explanation is written in black ink. Here, the theory part is written in black ink irrespective of whether the diagrams are drawn with a pencil or not. So, the two events are independent and hence the probabilities of occurrence of these two events are unconditional.
Conditional Probability Formula:
The formula for conditional probability is given as:
P(A/B) = \[\frac{N(A\cap B)}{N(B)}\]
In the above equation,
P (A | B) represents the probability of occurrence of event A when event B has already occurred
N (A ∩ B) is the number of favorable outcomes of the event common to both A and B
N (B) is the number of favorable outcomes of event B alone.
If ‘N’ is the total number of outcomes of both the events in a sample space S, then the probability of event B is given as:
P(B) = \[\frac{N(B)}{N}\] → (1)
Similarly, the probability of occurrence of event A and B simultaneously is given as:
P(A ∩ B) = \[\frac{N(A\cap B)}{N}\]→ (2)
Now, in the formula for conditional probability, if both numerator and denominator are divided by ‘N’, we get
P(A/B) = \[\frac{\frac{N(A\cap B)}{N}}{\frac{N(B)}{N}}\]
Substituting equations (1) and (2) in the above equation, we get
P(A/B) = \[\frac{P(A\cap B)}{P(B)}\]
Conditional Property Problems:
Question 1) When a fair die is rolled, find the probability of getting an odd number. Also find the probability of getting an odd number given that the number is less than or equal to 4.
Solution:
In the given questions there are two events. Let A and B represent the 2 events.
A = Getting an odd number when a fair die is rolled
B= Getting a number less than 4 when a fair die is rolled
The possible outcomes when a die is rolled are {1, 2, 3, 4, 5, 6}
The total number of possible outcomes in this event of rolling a die: N = 6
For the event A, the number of favorable outcomes: N (A) = 3
For the event B, the number of favorable outcomes: N (B) = 4
The number of outcomes common for both the events: N (A ∩ B) = 2
The probability of event A is given as:
P(A) = \[\frac{N(A)}{N} = \frac{3}{6}\] = 0.5
The probability of occurrence of event A given event B is
P(A/B) = \[\frac{N(A\cap B)}{N(B)} = \frac{2}{4}\] = 0.5.
Fun Facts:
The conditional probability of two events A and B when B has already occurred is represented as P (A | B) and is read as “the probability of A given B”.
The probability of occurrence of an event when the other event has already occurred is always greater than or equal to zero.
If the probability of occurrence of an event when the other event has already occurred is equal to 1, then both the events are identical.
FAQs on Conditional Probability and It's Examples
1. What is conditional probability and how does it differ from ordinary probability?
Conditional probability measures the likelihood of an event A occurring given that another event B has already taken place. This differs from ordinary probability, which examines the chance of an event happening independently. In conditional probability, the sample space is limited to outcomes where B occurs, making it distinct from unconditional or absolute probability.
2. Can you explain the formula for conditional probability with an example relevant to CBSE Class 12 Maths?
The formula for conditional probability is P(A | B) = P(A ∩ B) / P(B), where P(A | B) is the probability of event A occurring given that B has occurred. For example, if you roll a fair die and want the probability of getting an odd number (A) given that the number is less than 4 (B), A ∩ B = {1, 3}, N(A ∩ B) = 2, N(B) = 3, so P(A | B) = 2/3.
3. Why is conditional probability important in real-world situations?
Conditional probability models situations where outcomes depend on other events, helping in decision making under uncertainty. For instance, predicting weather conditions based on existing forecasts, or calculating the likelihood of a student passing an exam given that they attended classes regularly, both require analyzing how one event's outcome changes the probability of another.
4. What are independent events and how do they affect conditional probability calculations?
Two events are independent if the occurrence of one does not affect the probability of the other. In this case, P(A | B) = P(A). This is critical because, for independent events, conditional probability equals the original probability, showing there's no dependency or influence between events.
5. How do you identify mutually exclusive vs. independent events in probability problems?
In probability, mutually exclusive events cannot occur together (P(A ∩ B) = 0), so the occurrence of one prevents the other. Independent events can occur simultaneously and don't impact each other's probabilities. Recognizing the type of event is important to select the correct probability approach in exams.
6. What misconceptions do students often have about conditional probability?
Common misconceptions include:
- Assuming that P(A | B) is always equal to P(B | A)
- Ignoring changes in the sample space when conditioning on another event
- Overlooking the need to check if events are independent or dependent before calculating
7. How is conditional probability applied in board examination questions?
In the CBSE Class 12 Board exams, conditional probability questions typically involve scenarios such as drawing cards, tossing coins, or real-life problem statements. Success involves clearly identifying given and required events, constructing the correct sample space, and applying P(A | B) = P(A ∩ B) / P(B) using numerical values where appropriate.
8. What’s the significance of the sample space in solving conditional probability problems?
The sample space defines all possible outcomes for a random experiment. When calculating conditional probability, the sample space is restricted to the subset where the known event occurs. Correctly identifying the reduced sample space ensures accurate calculation of probabilities in exam scenarios.
9. What types of errors should students watch out for while answering conditional probability questions in exams?
Students should avoid the following errors:
- Misidentifying the conditioned event and thus the denominator
- Failing to reduce the sample space correctly
- Confusing independence with mutual exclusivity
- Incorrectly applying formulas or mixing up P(A | B) and P(B | A)
10. How could a change in one event influence the probability of another in a conditional probability scenario?
When events are dependent, the occurrence of one changes the likelihood of the other. For example, if a card is drawn from a deck and not replaced, the probability of a particular suite occurring next changes because the sample space is reduced by one card, altering the conditional probabilities accordingly.
