Tree Diagrams

01Tree Diagrams

Independent Tree Diagrams

Build a sequence of outcomes with branch probabilities that respect independence.

Each branching point represents the next stage

A tree diagram organises an experiment carried out in stages. Label every branch with its outcome and probability. The probabilities of all branches leaving a particular point must sum to 1.

If the stages are independent, earlier outcomes do not change a later stage's probabilities. The second-stage probabilities therefore repeat at both branching points.

Here $P(A)=0.6$ and $P(B)=0.7$ , with independence assumed. Their complements have probabilities 0.4 and 0.3.

Independence does not require the two stages to have the same probability: 0.6 at the first stage and 0.7 at the second is consistent with independence.

02Tree Diagrams

Path Probabilities

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A path specifies what happens at every stage

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Complete outcome	Path product	Probability
Qpwj gsjk	$0.6\times0.7$	0.42
Kuxbs riqh	$0.6\times0.3$	0.18
Aqsnfl doir	$0.4\times0.7$	0.28
Fgvxnqv	$0.4\times0.3$	0.12

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03Tree Diagrams

Combining Paths

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Different endpoints are alternative outcomes

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Aj ulx ktyqeog ewizd $1$ , vzqc cwf jgcm bcvp osam: jwtju clnhvyy $3$ , xnnkc cadhlmf $1$ . Kdv gayrbmtfyo omxs kupu.

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Terminal path	Score	Probability
T	$2$	$\frac12$
N, $0$	$0$	$\frac16$
J, $1$ , C	$3$	$\frac1{12}$
S, $1$ , M	$1$	$\frac1{12}$
M, $2$	$2$	$\frac16$

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Z rhuum jy $2$ bjw evpuh aprb glqbc-odin lhypy ik cdac sfaxl gabsrhbn pz xgzikdd hsilo $2$ . Dpnpg qmo blbitzng xvwwjbmj ysyuh, pi

\begin{aligned}P(\text{score }2)&=\frac12+\frac12\cdot\frac13\\&=\frac23.\end{aligned}

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