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Calculate the probability that both independent events A AND B occur together.
Two events A and B are independent if the occurrence of one event does not affect the probability of the other event. In other words, knowing that A happened tells you nothing about whether B will happen.
Only valid when A and B are independent!
Events A and B are independent if and only if: P(B|A) = P(B)
This means the probability of B is the same whether or not A occurred.
One event does not affect the other. Examples: coin flips, dice rolls, separate experiments.
One event affects the probability of the other. Example: drawing cards without replacement.
Putting items back after selection creates independence. The conditions reset each time.
Not putting items back creates dependence. The pool changes after each selection.
You flip two fair coins. What is the probability of getting Heads on both?
P(both Heads) = 1/4 = 25%
Assuming independence without checking - Always verify events are truly independent before multiplying!
Confusing with vs without replacement - Drawing without replacement creates dependence.
Gambler's fallacy - Past independent events do not affect future ones. After 5 heads, P(next head) is still 1/2!
The multiplication rule extends to any number of independent events:
P(rolling 6 five times in a row) = (1/6)⁵ = 1/7776 ≈ 0.013%