Mathematics · Grades 9–12
Probability: choose the rule and check its conditions
Do not assume independence or mutual exclusion unless justified.
Understand the idea
Complement
A and its complement cover all outcomes without overlap.
P(not A) = 1 − P(A)
P(A) = 0.3 gives P(not A) = 0.7.
Union
Subtract the overlap to avoid double counting.
P(A or B) = P(A) + P(B) − P(A and B)
0.4 + 0.5 − 0.2 = 0.7.
Conditional
Requires P(B) > 0. The reference sample is restricted to B.
P(A given B) = P(A and B)/P(B)
0.12/0.3 = 0.4.
Multiplication
Use the conditional probability when events may be dependent.
P(A and B) = P(A) × P(B given A)
0.5 × 0.4 = 0.2.
Independence
Use only when the independence model is appropriate.
P(A and B) = P(A)P(B)
Two independent fair coin heads: 1/2 × 1/2 = 1/4.
Expected count
An expected count is not a guaranteed actual count.
expected count = n × p
20 trials with p = 0.3 have expected count 6.
Further reading and review references
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Preview material · independent educational review required. Grade bands are approximate, not official curriculum alignment.