Chapter 15: Probability

15.1 The basics of probability

Alkana and her friends want to go to an outdoor music festival. They can choose between two music festivals, one in early September and one in late October. Because the festivals are outdoors, Alkana is worried about rain. To find out about the chance of rain for each festival, Alkana looks online to find out how much rain there was in her area for the last ten years. She looks up early September and late October. She finds out that there has been rain 8 times over the last 10 years in in early September, but only 3 times in late October. She now knows that the probability of rain in late October is far lower, she suggests to her friends that they attend the late October festival.

In the example above, we are talking about the probability of something happening. We often use the word "chance" for probability, as when Alkana finds out about the chance of rain for each festival.

probability Probability is a mathematical way of describing how likely an event is to occur.

Some things always happen, so we can say it is certain that they will happen.
For example, it is certain that Tuesday is followed by Wednesday.

Some things will never happen, so we can say they are impossible.
For example, it is impossible for Wednesday to be followed by Tuesday.

Some things are not certain, but are not impossible either. They may happen or they may not happen. We can say that they are unlikely or likely to happen.

Sometimes there is an equal chance or a 50% chance that one of two things could happen. For example, if you toss a coin it might land on heads, or it might land on tails.

These words that are used to describe different probabilities are shown on the probability scale below.

We can also use numbers between 0 and 1 to describe the probability of an event happening:

• An event that is impossible is given a probability of 0.
• An event that is certain is given a probability of 1.
• All other probabilities on this scale are between 0 and 1.
  • If there is a 50% chance, the fraction is $\frac{1}{2}$.
  • The less likely an event is to happen, the smaller the fraction.
  • The more likely an event is to happen, the larger the fraction.

Listing the outcomes of a probability experiment

We can use dice, coins and playing cards to do probability experiments. A probability experiment is any event that has a defined set of possible results or outcomes.

For example, all the possible outcomes of throwing a die are given in the set {1; 2; 3; 4; 5; 6}. All the possible outcomes of tossing a coin are given in the set {heads; tails}.

We have one die (singular) and many dice (plural).

Although a probability experiment has a defined set of possible outcomes, we don't know which one of these outcomes will take place each time. We perform an action, and record the outcome of the action each time.

probability experiment A probability experiment is an event that has a defined set of possible outcomes.

outcome An outcome is the possible result of an experiment.

Playing cards are often used to demonstrate probability experiments.

There are 52 cards in a pack of cards. These are divided into four suits: clubs, hearts, spades and diamonds.

In the illustration above:

• The red cards in the top row are the suit of hearts.
• The red cards in the second row are the suit of diamonds.
• The black cards in the third row are the suit of clubs.
• The black cards in the fourth row are the suit of spades.
• There are 13 cards in each suit.
• There are 26 black cards and 26 red cards in a pack of cards.
• There are 12 picture or face cards in a pack of cards (shown in the last three columns): four Jacks (J), four Queens (Q) and four Kings (K).
• There are four different Aces in a pack of cards, one for each suit. The Aces are shown in the first column.
• In each suit there are ten number cards. The Ace usually counts as the number 1, and then there are the cards numbered 2 to 10.
• There are therefore $4 \times 10 = 40$ number cards in a pack of cards.
• A pack of playing cards usually has two Joker cards, which are often removed for probability experiments.

15.2 Calculating probabilities

Probability gives us a mathematical way of working out how likely it is that an event will happen. To do this, we first need to be very clear about the particular outcome that we are looking for.

For example, if we are flipping a coin, we might want to calculate the probability of the coin landing on tails. Then "landing on tails" is the outcome we are looking for. We call this the favourable outcome.

To calculate the probability of an event, we look at the ratio of the number of favourable outcomes to the number of possible outcomes. The number of possible outcomes is the total number of outcomes that is possible for one event.

When we flip a coin, we know that the number of possible outcomes is 2, because the set of all the possible outcomes is { $\text{heads; tails}$}.

If the favourable outcome is "landing on tails", there is 1 favourable outcome. There are 2 possible outcomes. So, the ratio of the number of favourable outcomes to the number of possible outcomes is 1 : 2.

You have learnt that a ratio can be written as a fraction, so this ratio can also be written as $\dfrac{1}{2}$.

favourable outcome The favourable outcome is the particular outcome that we are looking for.

possible outcomes The total number of outcomes that is possible for one event is the possible outcomes.

To calculate the probability of an event happening, we can use this formula:

$\text{Probability of an event}$ = \(\dfrac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}\)

We can only use this formula to work out probability when all the outcomes of an activity are equally likely. For example, when we flip a coin, it is equally likely that we will get a heads or a tails, or when we toss a die, it is equally likely that we will get a 1, 2, 3, 4, 5 or 6.

To use this formula to calculate the probability of an event, we need to:

• find the number of possible outcomes
• find the number of favourable outcomes
• express the ratio as a fraction
• or we can divide the number of favourable outcomes by the number of possible outcomes in the fraction to get a decimal fraction.

Using shapes to calculate probabilities

We can investigate probabilities using a group of shapes. For the probability experiment, we will assume that the group of shapes is in a bag, so the person picking out a shape cannot see the shapes. The person will pick out one shape at a time.

We will calculate the probability of selecting or picking one shape from a group of shapes. To do this, we need to know:

• the total number of shapes we are picking from
• which shape we are choosing as the favourable outcome
• how many of the chosen shapes there are altogether in the bag.

When we have worked this out, we can substitute into the probability formula.

Using dice, coins and playing cards to calculate probabilities

When we do probability experiments using dice and coins, we assume that we have a fair dice and a fair coin.

• When a fair die is thrown, each of the numbers on the die is equally likely to occur.
• When a fair coin is flipped, the heads and the tails are equally likely to occur.

In a fair die, all the edges and corners are exactly the same, and the mass is evenly spread through the die. If the edges and corners were rubbed in different ways, or if there was a weight inside the die, the die would tend to fall on one number more than the others. This would be an unfair die.

A coin that lands on tails more often than on heads, because it is weighted or shaped in some way, would be an unfair coin.

15.3 Practical applications

We can determine the probability of events occurring in everyday life.

An example of one practical application is to use the information of how students travel to school each day to find the probability that one of the students, chosen at random, uses a particular method to get to school every day.

Often the information we are working with is given in a frequency table, which means that it is easy to determine the values we need to calculate probabilities.

15.4 Summary

• Probability is a mathematical way of describing how likely an event is to occur.
• We can use words to describe the probability of something happening.
  • If something always happens, we say it is certain.
  • If something never happens, we say it is impossible.
  • If something may or may not happen, we say it is likely, unlikely, or that there is a 50% or equal chance that it will happen.
• We can also use numbers from 0 to 1 to describe the probability of something happening.
  • An event that is impossible is given a probability of 0.
  • An event that is certain is given a probability of 1.
  • All other probabilities are fractions between 0 and 1.

• When all the outcomes of a probability experiment are equally likely, we can calculate the probability of one of the events using this formula:

  $\text{Probability of an event}$ = \(\dfrac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}\)

  • The favourable outcome is the particular outcome that we are looking for.
  • The total number of outcomes that is possible for one event is the possible outcomes.