Understanding the fundamentals of probability for combined events is crucial in statistics. This includes independent events, where probabilities are multiplied, and mutually exclusive events, where they are added. Conditional probability and probability trees are also discussed, providing insights into event dependencies and complex probability calculations. These concepts are key to solving a wide range of probability problems.
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Probability is a measure of the likelihood of an event's occurrence
Multiplication Rule
The multiplication rule states that the joint probability of two independent events is the product of their individual probabilities
Example of Independent Events
When flipping a coin and rolling a die, the probability of both events occurring is the product of their individual probabilities
Addition Rule
The addition rule states that the probability of either mutually exclusive event occurring is the sum of their individual probabilities
Example of Mutually Exclusive Events
When flipping a coin, the probability of getting either heads or tails is 1
Systematic listing of outcomes is a methodical approach to calculating the probability of combined events
The basic probability formula is the number of favorable outcomes for an event divided by the total number of possible outcomes
In a game with four cards, two of which are winners, listing the outcomes can help determine the probability of drawing a winning card
Conditional probability is the likelihood of an event occurring given that another event has already occurred
The formula for conditional probability is the joint probability of two events divided by the probability of the first event
If the probability of a person being a smoker is 0.2 and the probability of a smoker having lung cancer is 0.1, then the probability of a person having lung cancer given they are a smoker is 0.5
Probability trees are a graphical representation that helps in calculating conditional probabilities
Probability trees are particularly useful for visualizing the dependencies between events and calculating complex probabilities
A probability tree can be used to calculate the likelihood of a patient having a disease based on successive test results
00
In statistics, ______ measures how likely an event is to occur, and for combined events, they can be ______ or ______.
Probability
independent
mutually exclusive
01
The joint probability of two ______ events is found by multiplying their individual probabilities, expressed as ______.
independent
P(A∩B) = P(A) × P(B)
02
Definition of independent events in probability
Events where the occurrence of one does not affect the probability of the other.
