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More Tree Diagrams

More Probability Topics

* Example*

A coin and a die are thrown at random. Find the probability of:

a) getting a head and an even number

b) getting a head or tail and an odd number

* Solution: *

We can use a tree diagram to help list all the possible outcomes.

From the diagram, n(*S*) = 12

a) Let *A* denote the event of a head and an even number.

* A* = ((H, 2), (H, 4), (H, 6)} and n(A) = 3

b) Let *B* denote the event a head or tail and an odd number.

* B* = {(H, 1), (H, 3), (H, 5), (T, 1), (T, 3), (T, 5)}

* Example: *

Clare tossed a coin three times.

a) Draw a tree diagram to show all the possible outcomes.

b) Find the probability of getting:

(i) Three tails.

(ii) Exactly two heads.

(iii) At least two heads.

* Solution: *

a) A tree diagram of all possible outcomes.

b) The probability of getting:

(i) **Three tails.**

Let *S* be the sample space and *A* be the event of getting 3 tails.

n(*S*) = 8; n(*A*) = 1

P(*A*) =

ii) **Exactly two heads. **

Let *B* be the event of getting exactly 2 heads.

n(*B*) = 3

P(*B*) =

iii) **At least two heads**.

Let *C* be the event of getting at least two heads.

n(*C*) = 4

P(*C*) =

*Example: *

A spinner is labelled with three colors: Red, Green and Blue. Marcus spun the spinner once and tossed a coin once.

a) Draw a tree diagram to list all the possible outcomes.

b) Calculate the probability of getting blue on the spinner and head on the coin.

c) Calculate the probability of red or green on the spinner and tail on the coin.

* Solution: *

a) A tree diagram of all possible outcomes.

b) The probability of getting blue on the spinner and head on the coin.

Let *S* be the sample space and *A* be the event of getting blue and head

n(*S*) = 6 ; n(*A*) = 1

P(*A*) =

c) The probability of red or green on the spinner and tail on the coin.

Let *B* be the event of getting red or green and tail

n(*B*) = 2

P(*A*) =

Probability using Probability Trees.

Problem: A coin is biased so that it has 60% chance of landing on heads. If it is thrown three times, find the probability of getting:

(a) 3 heads

(b) 2 heads and a tail

(c) at least one head

The following video gives more examples of probability involving coins and using tree diagrams.

Sam tosses a biased coin twice. Given p(head) = 3/10. Find p(a head and a tail)

You flip 3 coins. What is the theoretical probability of getting 2 heads and 1 tails?

Sam tosses a biased coin twice. Given p(head) = 3/10. Find p(a head and a tail)

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