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IB DP Maths AA SL Study Notes

4.4.1 Basics of Conditional Probability

Introduction to Conditional Probability

In everyday life, we often make predictions based on conditions. For instance, the probability of it raining might be higher if there are dark clouds in the sky. Here, the event of it raining is dependent on the condition of there being dark clouds. This is the essence of conditional probability.

Definition

Conditional probability can be thought of as the probability of an event occurring, given that another event has already taken place. If we denote two events as A and B, the conditional probability of A occurring, given that B has already happened, is represented as P(A|B).

Formula and Explanation

The mathematical representation of conditional probability is:

P(A|B) = P(A and B) / P(B)

Where:

  • P(A|B) denotes the conditional probability of A given B.
  • P(A and B) represents the joint probability, which is the probability of both A and B occurring.
  • P(B) is the probability of event B.

For this formula to be valid, P(B) should not be zero because you cannot condition on an event that has zero probability.

To further elucidate, let's break down the formula:

The numerator, P(A and B), is the probability that both events A and B happen. The denominator, P(B), is the probability that event B happens. So, the formula essentially divides the likelihood of both events happening by the likelihood of the given event. This gives us a revised probability of event A in the context of event B.

Real-World Applications

Conditional probability isn't just a theoretical concept; it has practical implications in various domains:

1. Medicine: Doctors use conditional probability to determine the likelihood of a patient having a disease given a positive test result. This helps in making informed decisions about treatment.

2. Finance: In the stock market, traders might want to know the probability of a stock price rising given certain economic indicators. This aids in making investment decisions.

3. Weather Forecasting: Meteorologists use conditional probability to predict weather patterns based on current conditions.

4. Machine Learning: Algorithms, especially in classification tasks, often rely on conditional probability to make predictions based on given data.

Detailed Examples

Example 1: Medical Testing

Scenario: Consider a rare disease that affects 1% of a population. A test for this disease is 99% accurate. If a person tests positive, what's the probability they actually have the disease?

Solution: Let's denote:

  • D as the event of having the disease.
  • T as the event of testing positive.

We want to find P(D|T), the probability of having the disease given a positive test.

Using Bayes' theorem: P(D|T) = (P(T|D) * P(D)) / P(T)

Where:

  • P(T|D) is the probability of testing positive given that you have the disease, which is 0.99 (99% accurate test).
  • P(D) is the probability of having the disease, which is 0.01 (1% of the population).
  • P(T) is the total probability of testing positive.

The total probability of testing positive, P(T), can be found using the law of total probability: P(T) = P(T|D) * P(D) + P(T|¬D) * P(¬D) Where P(T|¬D) is the probability of testing positive given that you don't have the disease (false positive), and P(¬D) is the probability of not having the disease.

Plugging in the values, we can compute P(D|T).

Example 2: Card Game

Scenario: In a deck of cards, what's the probability of drawing a king, given that the card drawn is a face card?

Solution: There are 52 cards in a deck, with 12 face cards (4 each of Kings, Queens, and Jacks). Given that you've drawn a face card, there are 4 kings out of 12 face cards.

P(King|Face Card) = 4/12 = 1/3

So, the conditional probability of drawing a king given a face card is 1/3.

Key Points to Remember

  • Conditional probability offers a way to update our beliefs based on new evidence or information.
  • It's crucial for understanding complex probability topics and real-world applications.
  • The formula for conditional probability connects the joint probability of two events with their individual probabilities.

FAQ

Two events, A and B, are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. In terms of conditional probability, events A and B are independent if P(A|B) = P(A). This means that the probability of A occurring remains unchanged even if we know that B has occurred. If this condition is not met, then the events are dependent. Understanding the relationship between conditional probability and independence is vital, especially when dealing with multiple events in probability studies.

Conditional probability and joint probability are related but distinct concepts. Conditional probability, denoted as P(A|B), represents the probability of event A occurring given that event B has already taken place. On the other hand, joint probability, denoted as P(A and B), represents the probability of both events A and B occurring simultaneously. While conditional probability provides a revised probability based on a condition or prior knowledge, joint probability simply gives the likelihood of two events happening together without any conditions.

Conditioning on an event with zero probability is mathematically undefined. When calculating conditional probability using the formula P(A|B) = P(A and B) / P(B), if P(B) is zero, then the denominator becomes zero, leading to an undefined result. In practical terms, conditioning on an event with zero probability doesn't make sense. It's akin to asking the probability of an event A given an impossible condition. Thus, for conditional probability to be meaningful and well-defined, the conditioning event must have a non-zero probability.

Conditional probability is crucial in real-world scenarios because it allows us to make informed decisions based on prior knowledge or conditions. For instance, in medical fields, doctors use conditional probability to assess the likelihood of a patient having a disease given a particular test result. This helps in determining the best course of action for treatment. Similarly, in finance, investors use conditional probability to gauge the potential of an investment given certain economic indicators. Essentially, conditional probability provides a framework to update our beliefs or predictions based on new evidence, making it invaluable in various practical applications.

No, the value of conditional probability cannot be greater than 1. Probabilities, in general, always lie between 0 and 1, inclusive. A probability of 0 indicates that the event is impossible, while a probability of 1 indicates that the event is certain. Conditional probability, being a type of probability, adheres to this rule. If you ever compute a conditional probability that is greater than 1 or less than 0, it's likely that there's an error in the calculations.

Practice Questions

In a school, 60% of students play football, and 40% of students play basketball. If 30% of students play both football and basketball, what is the probability that a student chosen at random plays football given that they play basketball?

To solve this, we can use the formula for conditional probability:

P(F|B) = P(F and B) / P(B)

Where:

  • P(F|B) is the probability of playing football given playing basketball.
  • P(F and B) is the probability of playing both football and basketball, which is 30%.
  • P(B) is the probability of playing basketball, which is 40%.

Plugging in the values:

P(F|B) = 30% / 40% = 75%

So, the probability that a student plays football given that they play basketball is 75%.

In a city, the probability that it rains on any given day is 20%. If it rains, the probability that there will be traffic congestion is 70%. What is the probability that there is traffic congestion given that it rained?

This is a straightforward application of the formula for conditional probability. Given:

  • P(R) = Probability that it rains = 20%
  • P(T|R) = Probability of traffic congestion given that it rained = 70%

The probability that there is traffic congestion given that it rained is simply P(T|R), which is 70%. Thus, if it rained, there's a 70% chance of traffic congestion in the city.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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