What Are Independent Events?
Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
Think about tossing a coin and rolling a die at the same time. Whether the coin lands on heads or tails has absolutely no effect on which number comes up on the die — these are classic independent events.
Another everyday example: picking a ball from one box and then picking a ball from a completely separate box. What you get from the first box cannot influence what comes out of the second box.
Imagine you and a friend each flip your own separate coin at the same time. Your result (heads or tails) has zero influence on your friend's result. Each flip lives in its own little world — that's independence!
The Multiplication Rule for Independent Events
When two events A and B are independent, we can find the probability of both occurring by simply multiplying their individual probabilities together:
This rule extends to any number of independent events. For three independent events A, B, and C:
The key word here is "and" — you're looking for the probability that all the events happen together. Multiplying probabilities always makes the result smaller, which makes intuitive sense: getting multiple specific outcomes in a row is rarer than getting just one.
You can only use the multiplication rule when the events are truly independent. If one event affects the other (for example, drawing cards without replacing them), a different approach is needed.
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