# GCSE Maths Unit 1 - Tree Diagram

##### December 16, 2015

**GCSE Maths – Tree Diagrams**

A tree diagram is a diagram that is used to help us see all possible outcomes of an event and then calculate their probabilities. Each branch in a tree diagram represents a possible outcome and to find the probability of that particular outcome we multiply together all the individual probabilities along that branch.

**Example**

*A coin is tossed three times, falling either ‘Heads’ or ‘Tails’. Construct the tree diagram.*

**NB: **If the probabilities at each toss are P(head) = h, P(tail) = t [NB: t = 1-h]

Then... P(H,H,H) = h x h x h =h^{3}

P(H,H,T) = P(H,T,H) = P(T,H,H) = h x h x t =h^{2}t

P(T,T,H) = P(T,H,T) = P(H,T,T) = t x t x h = t ^{2}h

P(T,T,T) = t x t x t = t^{3 }

**Sample question**

*What are the probabilities of getting a) exactly 2 Heads and b) at least 1 Head *

*If i) the coin is fair [i.e. P(Head) = P(Tail) = 0.5]? ii) the coin is unfair, with P(Head) = 0.4?*

In part i) h = t = 0.5, and in part ii) h = 0.4, so t = 1 – 0.4 = 0.6, and in both parts:

a) P(exactly 2 Heads) = P(H,H,T) + P(H,T,H) + P(T,H,H) = 3 x P(H,H,T) =3h^{2}t

b) P(at least 1 Head) = 1 – P(no Heads) = 1 – P(3 Tails) = 1 - t ^{3}

i) a) P(2 Heads) = 3(0.5)^{2}(0.5) = 0.375, b) P(at least 1 Head) = 1 - (0.5)^{3} = 0.875

ii) a) P(2 Heads) = 3(0.4)^{2}(0.6) = 0.288, b) P(at least 1 Head) = 1 - (0.6)^{3} = 0.784

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