Tree Diagrams in Probability

Lessons

Senior 2 · Mathematics

Tree Diagrams in Probability

4 lessons

Lesson 1 of 4

The probability of an event

18 min

Why probability matters

"There is a 70% chance of rain in Musanze tomorrow." A farmer uses that sentence to decide whether to spread beans out to dry. A tea factory tests a sample of leaves to judge a whole delivery. Insurance companies, doctors and football coaches all use probability: a number that measures how likely something is. In this unit you will calculate probabilities, and then use tree diagrams to handle experiments with more than one stage.

The vocabulary

An experiment is an activity with an uncertain result, such as throwing a die or picking a bead from a bag.
An outcome is one possible result.
The sample space S is the set of all possible outcomes. For one throw of a die, S = {1, 2, 3, 4, 5, 6}.
An event is a set of outcomes we are interested in. "An even number" is the event E = {2, 4, 6}.

The probability of an event

When all the outcomes are equally likely:

P(E) = number of outcomes in E ÷ number of outcomes in S = n(E) ÷ n(S)

Worked example 1

A fair die is thrown. Find the probability of a prime number.
The primes on a die are 2, 3 and 5, so n(E) = 3 and n(S) = 6. P(prime) = 36 = 12.

Worked example 2

A bag holds 5 red, 3 green and 2 yellow beads, all the same size. One is picked without looking. Find P(green) and P(not green).
n(S) = 5 + 3 + 2 = 10. P(green) = 310. The beads that are not green number 7, so P(not green) = 710.

The probability scale

Every probability lies between 0 and 1: 0 ≤ P(E) ≤ 1.
P(E) = 0 means E is impossible (throwing a 7 on a die); P(E) = 1 means E is certain (throwing a number from 1 to 6).
The complement of E, written E′ ("not E"), contains every outcome not in E. Since one of them must happen, P(E′) = 1 − P(E). In Worked example 2: 1 − 310 = 710.

Probability from experiments

Sometimes outcomes are not equally likely, and we estimate a probability from what has happened before: the relative frequency. Worked example 3. A striker in a school team has scored 18 of her last 24 penalties. An estimate of the probability she scores the next one is 18 ÷ 24 = 0.75.

Tip

Count the sample space first. Write n(S) before anything else, and check every answer lies between 0 and 1. For "not" questions, use 1 − P(E); it is usually quicker than counting.

✦

Practice: try each one before you read the solution

1

A letter is chosen at random from the word RWANDA. Find P(A).

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Solution: 6 letters, 2 of them A: P(A) = 26 = 13.

2

A fair die is thrown. Find P(a number greater than 4).

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Solution: {5, 6}: 26 = 13.

3

Cards numbered 1 to 20 are shuffled and one is drawn. Find P(a multiple of 3) and P(not a multiple of 3).

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Solution: {3, 6, 9, 12, 15, 18}: 620 = 310. Not a multiple of 3: 1 − 310 = 710.

4

The probability of rain tomorrow is 0.65. What is the probability of no rain?

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Solution: 1 − 0.65 = 0.35.

Remember

For equally likely outcomes, P(E) = n(E) ÷ n(S).
0 ≤ P(E) ≤ 1: 0 is impossible, 1 is certain.
P(not E) = 1 − P(E).

Sources and credits

Introductory Statistics (OpenStax), 3.2 Terminology by Barbara Illowsky, Susan Dean and contributors, OpenStax, via LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.

The Rwandan examples, practice items and exercises are written by Ganzaa.

1Why probability matters

2The vocabulary

3The probability of an event

4The probability scale

5Probability from experiments

Practice: try each one before you read the solution

Why probability matters

The vocabulary

The probability of an event

The probability scale

Probability from experiments