Probability Basics: Rules and Concepts

Questions and Answers

If event A and event B are independent, what is the probability of both events occurring simultaneously?

$P(A \cap B) = P(A) \cdot P(B)$

Given two mutually exclusive events A and B, what is the probability of their union?

$P(A \cup B) = P(A) + P(B)$

If event A and event B are dependent, what is the probability of event B occurring given that event A has already occurred?

$P(B|A) = P(A \cap B) / P(A)$

If the probability of event A is 0.6 and the probability of event B is 0.4, what is the probability of at least one of the events occurring, assuming they are mutually exclusive?

0.76 Signup and view all the answers

If the probability of event A is 0.3 and the probability of event B is 0.5, what is the probability of both events occurring if they are independent?

$0.3 \cdot 0.5 = 0.15$ Signup and view all the answers

Given two events $A$ and $B$, what is the correct expression for the conditional probability of $A$ given $B$?

$P(A|B) = \frac{P(A \cap B)}{P(B)}$ Signup and view all the answers

In the coin flipping example, if the event $B$ is flipping one head and one tail, what is the conditional probability of flipping two heads given $B$?

$\frac{1}{3}$ Signup and view all the answers

If events $A$ and $B$ are mutually exclusive, what is the value of $P(A \cap B)$?

0 Signup and view all the answers

In the coin flipping example, if the total probability of all possible outcomes is 1, what is the value of $P(HH) + P(TT)$?

$\frac{1}{2}$ Signup and view all the answers

If events $A$ and $B$ are independent, which of the following statements is true?

All of the above Signup and view all the answers

Study Notes

Probability: Basic Concepts and Rules

Probability is a measure of uncertainty that allows us to estimate the likelihood of specific events or outcomes. It ranges from 0, meaning the event will never happen, to 1, indicating the event is guaranteed to occur. The study of probability involves various rules and concepts, including conditional probability and independence.

Probability Rules

There are several basic rules in probability theory that help us understand and compute probabilities:

Rule of Product (Multiplication Rule)

If A and B are events defined on a sample space, the probability of both A and B occurring simultaneously is: [ P(A\cap B) = P(A)\cdot P(B|A) ] This rule assumes that the events (A) and (B) are dependent, meaning that the occurrence of one event affects the probability of the other. If the events are independent, then (P(B|A)) becomes equal to (P(B)): [ P(A\cap B) = P(A)\cdot P(B) ]

Rule of Sum (Addition Rule)

If we have two mutually exclusive events (A_1) and (A_2), such that their union is the entire sample space: [ \Omega = A_1 \cup A_2 ] Their probabilities can be computed using the addition rule: [ P(\Omega) = P(A_1) + P(A_2) ] This rule assumes that the events (A_1) and (A_2) do not share any outcomes, so they cannot happen simultaneously.

Conditional Probability

Conditional probability deals with the likelihood of an event occurring given that another event has already happened. The conditional probability of an event (A) happening given that an event (B) has occurred is defined as: [ P(A|B) = \frac{P(A\cap B)}{P(B)} ] In other words, the conditional probability of (A) given (B) is the ratio of the joint probability of (A) and (B) to the probability of (B). This rule allows us to update our beliefs about the likelihood of an event based on new information. It's particularly useful when dealing with complex systems where multiple factors interact.

For example, let's consider a simple experiment involving flipping two coins. Let (A) be the event that both coins land heads up ((HH)), and (B) be the event that one coin lands heads up and the other lands tails up ((HT)). Since these two events cannot occur simultaneously, they are mutually exclusive. Using the addition rule, we can compute the total probability of all possible outcomes: [ P(A\cup B) = P(AHH) + P(AH) + P(HT) + P(TAH) + P(TT) = 1 ] Now, let's calculate the conditional probability of (A) given (B): [ P(A|B) = \frac{P(HH|HT)}{P(HT)} = \frac{\binom{2}{1}\binom{2}{1}}{\binom{4}{2}} = \frac{1}{3} ] So, if we know that one coin landed heads up, the probability of both landing heads up is (1/3).