Key Takeaways
- Conditional probability refers to the chances that some outcome (A) occurs given that another event (B) has also occurred.
- In probability, this is written as A given B, or as this formula: P(A|B), where the probability of A depends on that of B happening.
- Conditional probability can be contrasted with unconditional probability.
- Probabilities are classified as conditional, marginal (the base probability without any dependence on another event), or joint (the probability of two events occurring together).
- Bayes’ theorem is a mathematical formula used to calculate conditional probabilities.
Conditional probability can be contrasted with unconditional or “marginal” probability, which measures the chance of a single event without depending on any other. In contrast, conditional probability determines the likelihood of one event given that another event has occurred, linking them.
Understanding other terms for probabilities like independent probability looks at the probability of an event in isolation because it’s believed to be independent. Joint probability sums up the likelihood of two events happening together. Bayes’ Theorem then brings a sophisticated edge foreword by providing a way to reverse conditional probabilities mathematically.
Understanding Conditional Probability
Conditional probabilities are contingent on a previous event. They measure the likelihood of an event or outcome based on the occurrence of some earlier event.
Two events are said to be independent if one event occurring does not affect the probability that the other event will occur. However, if one event occurring or not does affect the likelihood that the other event will happen, the two events are said to be dependent. For example, a company’s stock price increasing after reporting higher-than-expected earnings.
Conditional probability is often written as the “probability of A given B” and notated as P(A|B). It is used in a variety of fields, such as insurance, economics, politics, and different fields of mathematics.
Conditional Probability Formula
P ( B | A ) = P ( A and B ) / P ( A )
Or:
P ( B | A ) = P ( A ∩ B ) / P ( A )
Where the letters are defined as follows:
- P = Probability
- A = Event A
- B = Event B
Unconditional probability, also known as marginal probability, measures the chance of something happening while ignoring any knowledge of previous or external events. Given this probability ignores new information, it remains constant.
Examples of Conditional Probability
Example 1: Marbles in a Bag
Step 1: Understand the scenario
You’re given a bag with six red marbles, three blue marbles, and one green marble. Thus, there are 10 marbles in the bag.
Step 2: Identify the Events
- Event A: Drawing a red marble from the bag
- Event B: Drawing a marble that is not green
Step 3: Calculate the probability of event B: P(B)
Event B is drawing a marble that is not green. There are 10 marbles altogether, nine of which are not green: the six red and three blue marbles.
P(B) = (Number of marbles that are not green)/(Total number of marbles) = 9/10
Step 4: Identify the intersection of events A and B: P(A∩B)
The intersection of events A and B involves drawing a red marble that is also not green. Since all red marbles are not green, the intersection is simple: the event of drawing a red marble.
Step 5: Calculate the probability of the intersection of events A and B: P(A∩B)
P(A∩B) = (Number of red marbles)/(Total number of marbles) = 6/10 = 3/5
Step 6: Calculate the conditional probability: P(A|B)
Using the formula,
P(A|B) = P(A∩B)/P(B) = (3/5)/(9/10) = 2/3
The conditional probability of drawing a red marble, given that the marble drawn is not green, is 2/3.
Example 2: Rolling a Fair Die
Step 1: Understand the scenario
You have a fair six-sided die. You want to determine the probability of rolling an even number, given that the number rolled is greater than four.
Step 2: Identify the events
The samples space for a six-sided die includes the numbers one through six. Thus,
- Event A: Rolling an even number {2, 4, 6}
- Event B: Rolling a number greater than four {5, 6}
Step 3: Calculate the probability of each event
P(A) - Probability of rolling an even number:
P(A) = Number of even numbers/Total possible outcomes = 3/6 = 1/2
P(B) - Probability of rolling a number greater than four:
P(B) = Number of outcomes greater than four/Total possible outcomes = 2/6 = 1/3
Step 4: Identify the intersection of events A and B
The intersection of these events includes the outcomes that satisfy both conditions…which is rolling a six.
Step 5: Calculate the probability of the intersection of events A and B
1P(A∩B) is the probability of rolling six: 1/6
Step 6: Calculate the conditional probability: P(B|A)
Using the formula,
P(B|A) = P(A∩B) / P(A) = (1/6)/(1/2) = 1/3
The probability that the number is even and also greater than four is 1/3.
Example 3: College Admission, Scholarship, and Stipend
Step 1: Understand the scenario
A student first wants to know the likelihood of being accepted to the university and then go on to apply/receive more – scholarsip/stipend.
Step 2: Define the Events
- Event A: Accepted to University
- Event B: Scholarship upon admission
- Event C: Stipend
Step 3: Probability Calculations
- Probability of Event A - Acceptance
1P(A) - University Acceptance = 100/1000 = 0.10.
- Probability of Event B - Scholarship upon Acceptance
1P(B|A) - Scholarship = 10/500 = 0.02.
- Probability of A and B - Scholarship while Accepted
1P(A∩B) = P(A) x P(B|A) = 0.10 x 0.02 = 0.002 or 2%.
- Probability of Scholarship leading if accepted and leading to Stipend
150% get stipend P(C|B) = 0.5.
- Given events for all – likelihood for all
1P(A∩B∩C) = P(A) x P(B∩C) x (CB)= 0.1 x 0.02x0.5 = 1/1%
2```.
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4## Conditional Probability vs. Other Types
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6### Conditional Probability:
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8A regular deck of cards can devine the different mutually and simultaneously occurring.
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10### Marginal Probability: …let's compute absolutely base probability without ethers.'
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12**Related Terms:** Joint probability, Marginal probability, Independent events, Bayesian statistics, Prior probability.
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15### References
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17## Get ready to put your knowledge to the test with this intriguing quiz!
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## What is conditional probability?
- [ ] The probability of an event occurring out of all possible outcomes
- [x] The probability of an event occurring given that another event has already occurred
- [ ] The probability of two events occurring together
- [ ] The probability of the occurrence of mutually exclusive events
## How is conditional probability usually denoted?
- [ ] P(A) - P(B)
- [ ] P(A) x P(B)
- [x] P(A|B)
- [ ] P(A) + P(B)
## Which of the following is the formula for conditional probability?
- [ ] P(A and B) / P(A)
- [ ] P(A) x P(B)
- [ ] P(A) - P(B)
- [x] P(A and B) / P(B)
## If event A has a probability of 0.4 and event B has a probability of 0.5, and the joint probability of A and B is 0.2, what is P(A|B)?
- [ ] 0.2
- [ ] 0.1
- [x] 0.4
- [ ] 0.5
## In the context of a deck of cards, if one card is drawn and it is a heart, what is the conditional probability that it is an Ace?
- [ ] 1/52
- [ ] 1/4
- [x] 1/13
- [ ] 1/2
## The law of total probability can be useful in calculating conditional probabilities. True or False?
- [x] True
- [ ] False
## Given two independent events A and B, what is P(A|B)?
- [x] P(A)
- [ ] P(B)
- [ ] P(A and B) / P(B)
- [ ] P(A) + P(B)
## Conditional probability can have values between which interval?
- [ ] 0 to 0.5
- [x] 0 to 1
- [ ] -1 to 1
- [ ] 1 to 2
## Why is conditional probability important in business decision making?
- [ ] It helps in avoiding market risks
- [ ] It ensures maximum profit
- [ ] It simplifies calculations
- [x] It helps in making informed decisions based on the occurrence of certain events
## Bayes' Theorem is related to:
- [ ] Absolute probability
- [ ] Frequentist probability
- [ ] The joint probability distribution
- [x] Conditional probability
