Probability

Probability

Probability is the mathematics of chance, of how likely something is to happen. Every probability sits on a scale from zero to one: zero means an outcome is impossible, one means it is certain, and everything else falls in between, with one half meaning an even chance. This module shows how to work out the probability of an event, how to combine the chances of two events, and the difference between events that cannot both happen and events that do not affect each other. It also covers relative frequency, where you estimate a chance by experiment, and expected frequency, where you predict how often something will happen over many tries. None of the arithmetic is hard once you are clear about what each idea means.

Words for probability

Learn these terms before the event types are matched up. They name the ideas used in probability.

Match each event type to its rule

Mutually exclusive against independent

Two ideas that students mix up are events that cannot happen together and events that do not affect each other. They use different rules.

What is the probability?

A fair six-sided dice is rolled once. What is the probability of rolling a number greater than four?

• One third, since two of the six numbers are greater than four
• One half
• One sixth
• Two

Relative frequency from an experiment

Sometimes you cannot work out a chance from theory, because the situation is not perfectly fair. Instead you run a trial many times and record what happens. The relative frequency is the number of times an outcome happened divided by the total number of tries. The more tries you do, the closer this estimate usually gets to the true value. So if a drawing pin lands point-up in sixty out of one hundred drops, the relative frequency of point-up is sixty hundredths, or six tenths. This is how you estimate a chance for a bent coin, a weighted spinner, or anything the theory cannot predict. More data gives a better estimate.

Pick the true probability facts

Select every statement about probability that is true.

• Probabilities run from zero to one
• A probability of zero means impossible
• A probability of one means certain
• Probabilities can be negative
• A probability can be bigger than one

Order the outcomes by how likely

Put these outcomes for an ordinary six-sided dice in order from least likely to most likely.

• Rolling a seven
• Rolling a six
• A tossed coin landing on heads
• Rolling a number greater than one
• Rolling a number less than seven

Complete the probability facts

The chance of an event happening, on a scale from zero to one, is its _____. Two events that cannot both happen at once are mutually _____, and for these you add the probabilities. Two events where one does not affect the other are _____, and for these you multiply the probabilities. The number of times an event is likely to happen in many trials is its _____ frequency.

Tap the impossible and the certain

Read these four outcomes for an ordinary six-sided dice. Tap the TWO that are either impossible or certain.

• rolling a seven
• rolling a number less than seven
• rolling a three
• rolling an even number

A dimmer switch from off to full

Think of chance as a dimmer switch running from off to full brightness. At one end the switch is fully off: the light can never come on, like an outcome that simply cannot happen. At the other end it is turned up all the way: the light is certainly on, like an outcome that is bound to happen. Most real situations sit somewhere in between, the light glowing at some level, neither dark nor dazzling. Learning to place an outcome on that dial, near the off end for unlikely, near the full end for near-certain, and in the middle for an even chance, is the heart of this whole topic. Once you can feel where an outcome sits, the numbers follow.

Work out the expected frequency

A spinner lands on red with a probability of 0.2. In 50 spins, how many times is it expected to land on red? Give the number only.

Reason through the probability

Read each situation and choose the best answer.

• You flip a fair coin twice. What is the chance of getting two heads?
• The probability that it rains tomorrow is 0.3. What is the probability that it does not rain?
• You want to estimate the chance a drawing pin lands point-up. What is the best method?

Build a probability point

Choose the word for each gap to complete one point about probability.

Explain probability

A friend finds probability confusing. Using what you have learned, explain the main ideas clearly.

• Explain the probability scale from zero to one
• Explain what mutually exclusive events are and how to combine them
• Explain what independent events are and how to combine them
• Explain what relative frequency is and when to use it
• Explain how to work out an expected frequency