Independent & repeated trials

Japanese school year: Math A

What you learn

This topic covers calculating probabilities for independent events and repeated trials under the same conditions. These principles are widely used to analyze repetitive processes, such as games and quality control. Prior knowledge of fundamental probability concepts and combinations is essential for this topic.

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Key points

This formula calculates the probability that both events $A$ and $B$ occur when they are independent and do not affect each other. Simply multiply their separate probabilities $P(A)$ and $P(B)$ .

P(A \cap B) = P(A)P(B)

This formula finds the probability that an event with success rate $p$ happens exactly $r$ times across $n$ repeated independent trials. Multiply the combination count ${}_n\mathrm{C}_r$ by the outcome probabilities.

P = {}_n\mathrm{C}_r p^r (1-p)^{n-r}

"Independent" means the outcome of one trial does not affect the other, while "mutually exclusive" means the two events cannot occur together. These are fundamentally different concepts.

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Practice 1Coming soon