How to Find the Probability of Compound Event
Read,3 minutes
A compound event consists of two or more simple events. To determine the probability of a compound event, we have to consider how the simple events are related.
Step 1: Determine If the Events are Independent or Dependent
Two events are independent if the occurrence of one event does not affect the occurrence of the other. Otherwise, they are dependent.
Step 2: Calculate the Probability for Independent Events
If \( A \) and \( B \) are independent events, the probability that both \( A \) and \( B \) occur is:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
Step 3: Calculate the Probability for Dependent Events
If \( A \) and \( B \) are dependent events, the probability that both \( A \) and \( B \) occur is:
\[ P(A \text{ and } B) = P(A) \times P(B|A) \]
Where \( P(B|A) \) is the probability of event \( B \) occurring given that event \( A \) has already occurred.
Step 4: Calculate the Probability of Either Event Occurring
If you want to find the probability of either \( A \) or \( B \) (or both) occurring, use:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Step 5: Consider Real-life Examples
Example: If you roll a fair six-sided die and flip a fair coin, what is the probability of rolling a 3 and getting a heads?
Since rolling a die and flipping a coin are independent events:
\[ P(3 \text{ and heads}) = P(3) \times P(\text{heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]
Step 6: Practice Regularly
Work on various examples and real-world scenarios to improve your understanding of compound probabilities.
Example
Let's consider a scenario involving a deck of cards and a die.
Scenario
You draw one card from a standard deck of 52 cards and roll a six-sided die simultaneously. What is the probability of drawing an Ace and rolling a 6?
Step 1: Determine Probabilities of Individual Events
1. The probability of drawing an Ace from a deck of cards: \( P(Ace) = \frac{4}{52} = \frac{1}{13} \)
2. The probability of rolling a 6 on a die: \( P(6) = \frac{1}{6} \)
Step 2: Calculate the Compound Probability
Since drawing a card and rolling a die are independent events, the combined probability is:
\[ P(Ace \text{ and } 6) = P(Ace) \times P(6) = \frac{1}{13} \times \frac{1}{6} = \frac{1}{78} \]
Conclusion
The probability of drawing an Ace from a standard deck of cards and rolling a 6 on a six-sided die simultaneously is \( \frac{1}{78} \).
