BackConditional Probability and the Multiplication Rule
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Conditional Probability and the Multiplication Rule
Conditional Probability
Conditional probability refers to the probability of an event occurring given that another event has already occurred. This concept is fundamental in understanding how probabilities change when additional information is known.
Notation: The conditional probability of event B given event A is denoted as .
Formula:
Interpretation: This formula calculates the probability that both events A and B occur, divided by the probability that event A occurs.
Independent and Dependent Events
Events are classified as independent or dependent based on whether the occurrence of one affects the probability of the other.
Independent Events: The occurrence of one event does not affect the probability of the other. For independent events, .
Dependent Events: The occurrence of one event affects the probability of the other. For dependent events, .
Example: Classifying Events as Independent or Dependent
Example 1: Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B). Solution: These events are independent because the outcome of the coin toss does not affect the outcome of the die roll.
Example 2: Driving over 85 miles per hour (A), and then getting in a car accident (B). Solution: These events are dependent because driving at high speed increases the chance of being in an accident.
The Multiplication Rule
The multiplication rule is used to find the probability that two events, A and B, both occur in sequence.
General Multiplication Rule:
For Independent Events:
This rule can be extended for any number of independent events.
Example: Using the Multiplication Rule to Find Probabilities
Example: Two cards are selected, without replacing the first card, from a standard deck of 52 playing cards. Find the probability of selecting a king and then selecting a queen. Solution: Since the first card is not replaced, the events are dependent.
Probability of selecting a king first:
Probability of selecting a queen given a king was already drawn:
Combined probability:
