Probability, counting, and trees

Updated 11 Oct 2026

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Sample spaces and events
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  Sample space $\Omega$ -> set of possible outcomes.
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  Event $A$ -> subset of $\Omega$ .
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  Probability bounds -> $0\leq P(A)\leq1$ .
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  Certain event -> $P(\Omega)=1$ .
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  Complement -> $P(A^c)=1-P(A)$ .
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  Finite equally likely outcomes -> $P(A)=\frac{|A|}{|\Omega|}$ .
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    counting favorable outcomes requires equally likely elementary outcomes.
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Addition rule
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  $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ .
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  Three events -> add individual probabilities -> subtract pairwise intersections -> add the triple intersection.
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Counting
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  the multiplication principle counts sequential choices.
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  permutations count ordered selections.
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  Combinations -> count unordered selections: $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ .
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  Reading the prompt -> “choose any three” usually means an unordered selection; treat it as ordered when order changes the outcome.
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Conditional probability and independence
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  $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ , provided $P(B)>0$ .
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  Independence -> $P(A\cap B)=P(A)P(B)$ .
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    observing one event does not change the probability of the other.
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  Mutually exclusive -> events cannot occur together.
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    if both have positive probability, they are not independent.
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  Without replacement -> usually changes the next conditional probability.
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Probability tree
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  Branch label -> conditional probability.
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  At each node -> outgoing branch probabilities sum to $1$ .
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  Path probability -> multiply along the path.
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  Event probability -> add mutually exclusive terminal paths.
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  Add a two-draw tree for with/without replacement; annotate path multiplication and the total probability of drawing different colors.
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Worked draw example
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  Draw twice from 3 red and 7 blue objects.
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  With replacement
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    $P(RR)=0.09$ .
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    $P(RB)=0.21$ and $P(BR)=0.21$ .
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    $P(BB)=0.49$ .
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    Different colors -> $0.42$ .
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  Without replacement
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    $P(RR)=\frac{1}{15}$ .
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    $P(RB)=\frac{7}{30}$ and $P(BR)=\frac{7}{30}$ .
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    $P(BB)=\frac{7}{15}$ .
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    Different colors -> $\frac{7}{15}$ .

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Probability, counting, and trees

Sample spaces and events

Addition rule

Counting

Conditional probability and independence

Probability tree

Worked draw example

Linked references 1