164. Introduction to Probability

Learning Intentions

• Understand that a probability is a number between 0 and 1, representing the likelihood of an event
• Know the meaning of the terms experiment, trial, outcome, event, sample space and complement
• Calculate the probability of simple events

Pre-requisite Summary

• Fractions can be used to Describe part of a whole
• A number line can show values between and
• Some events are impossible, some are certain, and others lie in between
• A list of possible results can be made for a simple chance situation
• Division can be used to compare favourable outcomes with total outcomes
• Simple fractions can be interpreted as probabilities

Worked Examples

Worked Example 1

State whether each probability is impossible, unlikely, even chance, likely or certain.

a)

b)

c)

Worked Example 2

For rolling a fair six-sided die, Identify the experiment, one trial, the sample space and the event “rolling an even number”.

Worked Example 3

A fair coin is tossed once.

a) Write the sample space.

b) Solve the probability of getting heads.

c) Find the probability of not getting heads.

Worked Example 4

A fair six-sided die is rolled once.

a) Find the probability of rolling a .

b) Find the probability of rolling a number greater than .

Worked Example 5

A bag contains red counters and blue counters. One counter is chosen at random.

a) Find the probability of choosing a red counter.

b) Find the probability of choosing a blue counter.

c) Find the probability of not choosing a red counter.

Worked Example 6

A spinner has equal sections: green, yellow, blue and red.

a) Find the probability of landing on yellow.

b) Find the probability of not landing on yellow.

Problems

Problem 1

State whether each probability is impossible, unlikely, even chance, likely or certain.

a)

b)

c)

Problem 2

For choosing a card numbered to at random, identify the experiment, one trial, the sample space and the event “choosing an odd number”.

Problem 3

A fair coin is tossed once.

a) Write the sample space.

b) Find the probability of getting tails.

c) Find the probability of not getting tails.

Problem 4

A fair six-sided die is rolled once.

a) Find the probability of rolling a .

b) Find the probability of rolling a number less than .

Problem 5

A bag contains green counters and white counter. One counter is chosen at random.

a) Find the probability of choosing a green counter.

b) Find the probability of choosing a white counter.

c) Find the probability of not choosing a green counter.

Problem 6

A spinner has equal sections: red, blue, orange and purple.

a) Find the probability of landing on orange.

b) Find the probability of not landing on orange.

Exercises

Understanding and Fluency

Exercise 1.

State whether each probability is impossible, unlikely, even chance, likely or certain.

a)

b)

c)

d)

e)

Exercise 2.

Write the matching probability for each description.

a) impossible

b) certain

c) even chance

Exercise 3.

For each situation, name the experiment and one possible outcome.

a) tossing a coin

b) rolling a six-sided die

c) choosing a coloured counter from a bag

Exercise 4.

Write the sample space for each experiment.

a) tossing a fair coin once

b) rolling a fair six-sided die once

c) choosing a card numbered to at random

Exercise 5.

Find the probability of each event.

a) rolling a on a fair six-sided die

b) tossing heads on a fair coin

c) choosing the card numbered from cards numbered to

Exercise 6.

Find the probability of each event.

a) rolling an even number on a fair six-sided die

b) rolling a number greater than on a fair six-sided die

c) choosing an odd number from cards numbered to

Exercise 7.

A bag contains red counters and blue counters. One counter is chosen at random.

a) Find

b) Find

c) Find

Exercise 8.

A spinner has equal sections: green, yellow, blue and red.

a) Find

b) Find

c) Find

Exercise 9.

For each event, find the complement probability.

a) for a fair coin

b) on a fair die

c) from a bag with red and black counters

Exercise 10.

A bag contains white, black and grey counter. One counter is chosen at random.

a) Find

b) Find

c) Find

d) Find

Reasoning

Exercise 11.

Explain why every probability must be between and .

Exercise 12.

A student says that the probability of an event can be . Explain why this is impossible.

Exercise 13.

Explain the difference between an outcome and an event.

Exercise 14.

Explain why the complement of an event has probability .

Exercise 15.

A student says the sample space for rolling a die is . Explain the error.

Problem-solving

Exercise 16.

A game uses a fair spinner with equal sections labelled , , and .

a) Write the sample space.

b) Find the probability of landing on .

c) Find the probability of not landing on .

Exercise 17.

A box contains pencils: red, blue and green. One pencil is chosen at random.

a) Find the probability of choosing a blue pencil.

b) Find the probability of not choosing a blue pencil.

c) Which colour is most likely to be chosen?

Exercise 18.

A fair die is rolled once.

a) Find the probability of rolling a number less than .

b) Find the probability of rolling a number not less than .

c) Describe the second event as a complement.

Exercise 19.

A card is chosen at random from cards numbered to .

a) Find the probability of choosing a multiple of .

b) Find the probability of not choosing a multiple of .

c) State whether choosing a multiple of is unlikely, even chance or likely.

Exercise 20.

In a lucky dip, there are prizes and are gold prizes. One prize is chosen at random.

a) Find the probability of getting a gold prize.

b) Find the probability of not getting a gold prize.

c) Explain whether getting a gold prize is likely or unlikely.

Potential Misunderstandings

• Thinking probability can be less than or greater than
• Confusing an event with a single outcome
• Thinking a trial means the same as the whole experiment
• Forgetting that the sample space includes all possible outcomes
• Believing the complement of an event is the same event
• Adding favourable outcomes incorrectly when finding probability
• Use the number of favourable outcomes as the denominator instead of the total number of outcomes
• Thinking all events on a die or spinner are equally likely even when the sections or contents are not equal
• Forgetting that impossible events have probability and certain events have probability
• Confusing “not” an event with only one other outcome instead of all remaining outcomes

Next: 165. Two-Step Probability Experiments