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The concept of independent events and the use of tree diagrams stem from early probability theory, which began in the 17th century. Mathematicians like Blaise Pascal and Pierre de Fermat laid the foundation for probability theory in correspondence about games of chance. Later, Jacob Bernoulli formalized many principles in his work Ars Conjectandi, while later developments by Andrey Kolmogorov provided a rigorous framework for probability.
1. Independent Events
Two events are independent if the occurrence of one event does not affect the probability of the other.
Definition:
Events A and B are independent if:
P(A∩B)=P(A)×P(B)
Examples:
Flipping a coin twice. The outcome of the first flip doesn't affect the second.
Rolling two dice. The result of one die doesn't impact the other.
Characteristics:
If P(A∣B)=P(A), then events A and B are independent.
The complement of an independent event may or may not be independent.
2. Dependent Events
Events are dependent if the outcome of one affects the probability of the other.
Example: Drawing two cards from a deck without replacement.
A tree diagram is a visual representation of all possible outcomes of events, structured in branches. Each branch shows an event and its probability.
How to Construct a Tree Diagram:
Start with a root: Represent the first event and its possible outcomes.
Branch out: For each possible outcome of the first event, draw branches for the second event.
Label probabilities: Add probabilities to each branch.
Calculate outcomes: Multiply probabilities along each branch to find the combined probabilities.
Example:
A bag contains 3 red and 2 blue balls. You pick two balls without replacement.
First event: pick a red or blue ball.
Second event: pick another red or blue ball.
Multiplication Rule for Independent Events:
P(A∩B)=P(A)×P(B)
Multiplication Rule for Dependent Events:
P(A∩B)=P(A)×P(B∣A)
Tree Diagram Probabilities:
Probability of a path=P(First event)×P(Subsequent events)
Independent Events and Mutually Exclusive Events are not the same thing. They are distinct concepts in probability, with key differences:
Two events are independent if the occurrence of one event does not affect the occurrence of the other.
Key Characteristics:
The probability of both events happening is the product of their individual probabilities: P(A∩B)=P(A)×P(B)
One event gives no information about the likelihood of the other.
Example:
Flipping a coin and rolling a die.
The outcome of the coin flip does not affect the number rolled on the die.
Two events are mutually exclusive if they cannot occur at the same time.
Key Characteristics:
If events AA and BB are mutually exclusive, then:P(A∩B)=0
If one event happens, the other cannot happen.
Example:
Rolling a single die: getting a 3 and getting a 5 are mutually exclusive events because a single die roll cannot be both 3 and 5 simultaneously.
No, mutually exclusive events can never be independent (except in trivial cases where one of the events has a probability of 0). If events are mutually exclusive, the occurrence of one implies the other cannot occur, so they are inherently dependent.