Probability · Ordinary Level
Tree Diagrams
Ordinary Level · Class 7 · Tap NEXT to begin
Section 1 of 2
Tree Diagrams
Branch by branch
A tree diagram shows every path. Multiply along the branches for each outcome; add the outcomes you want.
$5$ blue and $3$ green marbles ($8$ total). One is taken and replaced, then a second.
Worked example — both, only one, at least one
Find (i) $P(\text{both blue})$; (ii) $P(\text{only one blue})$; (iii) $P(\text{at least one blue})$.
$BB = \tfrac{5}{8}\times\tfrac{5}{8} = \tfrac{25}{64}$; $GG = \tfrac{3}{8}\times\tfrac{3}{8} = \tfrac{9}{64}$; $BG = GB = \tfrac{15}{64}$
(i) $\tfrac{25}{64}$
(ii) $BG$ or $GB = \tfrac{15}{64}+\tfrac{15}{64} = \tfrac{30}{64} = \tfrac{15}{32}$
(iii) $1 - P(\text{no blue}) = 1 - \tfrac{9}{64} = \tfrac{55}{64}$
$\tfrac{25}{64},\ \tfrac{15}{32},\ \tfrac{55}{64}$
Worked example — winning a game
The chance I win a game is $\tfrac{3}{5}$. Using a tree, in $2$ games find $P(\text{I win once})$.
$WW = \tfrac{9}{25}$; $WL = LW = \tfrac{6}{25}$; $LL = \tfrac{4}{25}$
Win once $= WL$ or $LW = \tfrac{6}{25} + \tfrac{6}{25} = \tfrac{12}{25}$
$\tfrac{12}{25}$
Section 2 of 2
Trees with Decimals
Worked example — Mary’s tests
$P(\text{Mary passes maths}) = 0.8$ and $P(\text{passes English}) = 0.7$. Find (i) $P(\text{passes both})$; (ii) $P(\text{passes one only})$.
$PP = 0.8\times0.7 = 0.56$; $PF = 0.24$; $FP = 0.14$; $FF = 0.06$
(i) $0.56$
(ii) $PF$ or $FP = 0.24 + 0.14 = 0.38$
$0.56,\ 0.38$
You try
$P(\text{pass theory test}) = 0.9$ and $P(\text{pass driving}) = 0.6$. Find $P(\text{passes both})$.
Multiply along the pass-pass branch.
$P(\text{both}) = 0.9 \times 0.6 = 0.54$
$0.54$
You try
$5$ blue and $8$ green marbles, one taken and replaced then a second. Find $P(\text{both green})$.
$P(\text{green}) = \tfrac{8}{13}$; multiply along green-green.
$\tfrac{8}{13} \times \tfrac{8}{13} = \tfrac{64}{169}$
$\tfrac{64}{169}$
That’s Class 7.
Tree diagrams — multiply along, add between — and ‘at least one’ via $1 - $ the opposite. Class 8: Venn diagrams.