Probability in mathematics means to calculate the likelihood of an event taking place. It is used in science, finance and our day-to-day decision-making. It's an important part of the GCSE Maths curriculum because it helps when tackling more advanced statistical concepts.
This GCSE Maths revision guide will explain basic probability, including how probability can be calculated and how to use it in common problems. If you need support revising this topic, consider finding a GCSE Maths tutor with TeachTutti.
What is Probability in Maths?
Probability is a branch of mathematics where we find the chance of an event happening. The probability of occurring events ranges from simple scenarios like rolling a die to complex situations like predicting weather patterns.
Definition and formula: Probability means how likely an event is to happen. It is expressed as a number between 0 and 1. If the probability of an event happening is 0, it won't happen, while an event with a probability of 1 is certain to take place.
The formula to find the probability is:
Probability (P) = Number of Favorable Outcomes / Total Number of Outcomes
For example, let's say there are four pairs of socks in a drawer and each pair is either red, green, blue or yellow. The probability of pulling out a pair of socks in the colour red is found by working out the number of favourable outcomes (red pair of socks) and dividing it by the total number of outcomes (4), which is 1/4.
Key terms:
- Experiment: An action or a process that have a clear set of potential outcomes. For example, flipping a coin or drawing a card from a deck.
- Outcome: The result of an experiment, such as getting heads when flipping a coin. This is a single run of the experiment.
- Event: Two or more outcomes of an experiment. If we continue the event of flipping a coin, the outcomes of heads, tails, tails, heads would be an event.
Examples in daily life:
- Weather forecasting: Probability can determine the likelihood of rain, storms, or sunshine and is used by meteorologists.
- Games of chance: Your odds of winning in games like poker, roulette and lotteries if you understand probability. For example, your probability of winning a lottery, however slight, is higher if you bet in bulk on a single occasion rather than a small amount regularly.
- Risk assessment: Probability is used by insurance companies to assess risk and determine premiums for customers.
1
What is the probability of rolling a 3 on a die with six sides?
Key Terminology in Probability
There are certain terms you'll often come across when studying and applying probabilities. Before we look in more detail at the topic, we'll cover the most important terms below:
- Experiment: An experiment is a process or action that returns a clear set of results. An example is flipping a coin or conducting a survey.
- Outcome: This is a single result of an experiment. For example, if we flip a coin and get heads, this is an outcome. Possible outcomes for rolling a die include 1, 2, 3, 4, 5, or 6.
- Sample space: All the possible outcomes of an experiment are called the sample space. A simple example is the same space for flipping a coin, which can be Heads or Tails.
- Event: An event is a set of multiple outcomes for the sample space. An event can be simple with a single outcome, or compound and have multiple outcomes. An example is rolling an even number on a die (2, 4, or 6).
- Mutually exclusive events: These are events that can't happen at the same time. For example, if you flip a coin, you can't get heads and tails at the same time.
- Independent events: An independent event isn't affected by the outcome of another event. For example, if we toss a coin twice, the result of the first toss doesn't influence the second toss (even though we think it does!)
- Complementary events: Complimentary events are all the outcomes in the sample space that aren't included in the event e.g. if event A is rolling a 3 on a die, the complement of event A is rolling 1, 2, 4, 5, or 6.
2
Which is the sample space for drawing a card from a standard deck of 52 cards?
Types of Probability
There are various types of probability you need to know for GCSE Maths. Here are the main types:
1. Theoretical probability: This is calculated based on the possible outcomes in a perfect world scenario. It's theoretical and you don't conduct any actual experiments. This is the formula:
P(A) = Number of Favourable Outcomes / Total Number of Possible Outcomes.
If we calculate the probability of rolling a 4 on a six-sided die, we get: P(4) = 1 / 6.
2. Experimental probability: This is also known as empirical probability. It's when you conduct an experiment and record the outcomes. This is the formula:
P(A) = Number of Times Event A Occurs / Total Number of Trials.
If you roll a die 60 times and get a 4 ten times, the experimental probability for rolling a 4 is: P(4) = 10 / 60 = 1 / 6.
3. Subjective probability: Subjective probability is when you use personal judgment, intuition, or experience to find the probability. It's used when the other types of probability are less effective, like predicting the outcome of a sports game.
4. Axiomatic probability: This probability is based on a set of axioms or rules. The most common axioms used in probability are those by Andrey Kolmogorov. They include:
- The probability of any event is a non-negative number.
- The probability of the sample space is 1.
- If two events in probability are mutually exclusive, the probability of either event occurring is the sum of the probabilities.
3
What probability is used when predicting the result of a football match based on a fan's intuition?
How to Calculate Probability
This is a step-by-step guide for calculating probability. Examples are included.
- Find the total number of outcomes: Determine the total number of possible outcomes for the experiment. This is your denominator in the probability formula.
- Work out the number of favourable outcomes: Determine how many of those outcomes are favourable to the event. This is your numerator in the probability formula.
- Apply the probability formula: P(A) = Number of Favorable Outcomes / Total Number of Possible Outcomes.
Examples:
- Tossing a Coin - There are two outcomes when flipping a coin: heads or tails. The probability of getting heads is: P(Heads) = 1 / 2
- Rolling a Die - There are six outcomes when rolling a six-sided dice. The chance of rolling a 5 is: P(5) = 1/ 6.
- Drawing a Card - There are four aces in a standard deck of 52 cards. The likelihood of drawing an ace is: P(Ace) = 4 / 52 = 1 / 13
Complex probability:
Drawing two cards - If you draw two cards and don't replace them, the probability will change after your first draw. For instance, the probability of drawing two aces back-to-back:
- The probability of drawing the first ace: 4 / 52 = 1 / 13.
- The probability of drawing the second ace after drawing the first: 3 / 51.
The combined probability is: P(Two Aces) = 4 / 52 × 3 / 51 = 1 / 13 × 1 / 17 = 1 / 221
Common Probability examples
A good way to firm up your understanding of probability is through practice. We've listed some of the types of probability problems that you may see in your GCSE Maths. Each problem includes our workings to find the solution:
1. Single event probability: How likely are you to draw a king from a standard deck of 52 cards?
- Number of favourable outcomes (drawing a king) = 4 (spade, diamond, club, heart)
- Total number of possible outcomes = 52
- Probability P(King) = 4 / 52 = 1 / 13
2. Multiple events probability: What is the probability of rolling an even number on a six-sided die?
- Number of favourable outcomes = 3 (2, 4, or 6).
- Total number of possible outcomes = 6.
- Probability: P(Even) = 3 / 6 = 1/2.
3. Combined events probability: What's the probability of drawing an ace and a king back-to-back from a deck of 52 cards?
- The probability of drawing an ace first: 4 / 52 = 1 / 13.
- The probability of drawing a king second with one card fewer: 4 / 51.
- Combined probability P(Ace and King)= 1/ 13 × 4 / 51 = 4 / 663.
4. Dependent events probability: What is the probability of picking a yellow card one after the other from a deck of 52 cards (26 yellow, 26 black)?
- The probability of drawing the first yellow card: 26 / 52 = 1/2.
- The probability of drawing the second yellow card with one fewer card: 25 / 51.
- Combined probability: P(Two yellows) = 12 × 25 / 51 = 25 / 102.
4
If you roll two six-sided dice together, what is the probability that the sum is 8?
Applications of probability
Here are some examples of how probability is used in our daily life:
- Weather forecasting: When meteorologists forecast the weather, they use probability to predict weather conditions based on weather models and historical data. For example, if a report says there's a 45% chance of hail, this means that this was the likelihood of hail on days in the past that match the current day.
- Insurance: Premiums and calculates and risks assessed using probability. Insurance companies will analyse data on accidents, illnesses and other metrics to work out the likelihood of an event happening and then set their rates accordingly.
- Games of chance: To achieve success in games like poker, you need a good grasp of probability to help you understand the odds and make informed decisions and strategies. For example, if you divide the total number of winning tickets by how many have been sold, you'll work out your chance of winning the lottery on a particular weekend.
- Finance and investment: Investors work out potential risks and returns for investment using probability. Models are used to predict stock market trends as well as interest rates and economic changes to name a few.
- Medicine: Medical professionals use probability to decide what treatment will be effective and the likelihood of contracting a disease. Clinical trials often use probability to find out the chances of side effects.
5
What does it mean if the weather forecast says there's a 30% chance of rain?
Conditional Probability and Bayes' Probability theory
So far, probability has been fairly straightforward. However, the areas of Conditional Probability and Bayes' Theorem are more complex. We use these topics to understand the independence of events and to make informed predictions using existing data:
1. Conditional probability: This is the likelihood that an event will happen given another event has already taken place. It's written as P(A|B), which means "the probability of A given B." The formula is P(A|B) = P(A intersection B). Breaking this down, P(A intersection B) is the probability of both events A and B happening and P(B) is the probability of event B only.
Example: What's the probability of drawing an ace in a deck of 52 cards if the first card was a king and it wasn't replaced?
- Number of aces = 4.
- Number of remaining cards after drawing a king = 51.
- P(Ace | King drawn first) = 4 / 51.
2. Bayes' theorem: The theorem by Bayes updates the probability of an event using new evidence. It is particularly useful in medical testing, where it updates the likely of a condition given a recent test result. The formula is P(A|B) = P(B|A) . P(A) / P(B):
- P(A|B) is the probability of event A given B is true.
- P(B|A) is the probability of event B given A is true.
- P(A) is the probability of event A.
- P(B) is the probability of event B.
Example: 1% of the population has a particular disease and a test for the disease has a 99% accuracy rate. What's the probability a person has the disease when they test positive?
- P(Disease) = 0.01
- P(Positive Test | Disease) = 0.99
- P(Positive Test | No Disease) = 0.01
- P(No Disease) = 0.99
Using Bayes' Theorem:
- P(Disease | Positive Test) = P(Positive Test | Disease) . P(Disease) / P(Positive Test)
- P(Positive Test) = (P(Positive Test | Disease) . P(Disease)) + (P(Positive Test | No Disease) . P(No Disease))
- P(Positive Test) = (0.99 . 0.01) + (0.01 . 0.99) = 0.0198
- P(Disease | Positive Test) = 0.99 . 0.01 / 0.0198 = 0.5
6
If a spade is drawn from a standard deck, what is the probability that it's a queen?
Conclusion
Probability is an important topic for GCSE Maths because it is the basis for advanced statistical concepts and real-world applications. If you want to explore the topic further, have a look at Statology's article on real-life examples of probability and Unacademy's exploration of probability in everyday life. Both resources give examples and applications so you can improve your understanding and appreciation for probabilities.
If you need further support in getting the hang of the topic, including advanced topics like a probability tree diagram, consider finding a GCSE Maths tutor with TeachTutti.
Frequently asked questions
Probability is a measure of the likelihood that an event will occur. We use numbers to represent this from 0 to 1. 0 means the event won't happen and 1 means it is certain to happen.
The use of probability is calculated using the formula: P(A)=Number of Favorable Outcomes / Total Number of Possible Outcomes.
There are several types in the field of probability, including theoretical, experimental, subjective and axiomatic probability.
This is how likely an event will happen given that another event has already occurred. The calculation is: P(A|B) = P(A intersection B) / P(B).
We see probability everywhere in day-to-day life. Among others, it's used in weather forecasting, insurance, finance, medicine and games of chance to predict outcomes and make informed decisions.
Glossary
- Probability - The meaning of probability is the likelihood that an event will occur. It is a numerical value from 0 (impossible) to 1 (certain).
- Experiment - An action or process that leads to a set of results, such as rolling a die.
- Outcome - The result of a single trial of an experiment, such as flipping heads in a coin toss.
- Sample space - All possible outcomes for an experiment. For instance, the sample space for a coin toss is Heads and Tails.
- Event - A set of multiple outcomes. For example, a set for flipping a coin could be: heads, tails, heads, heads.
- Mutually exclusive events - Events that cannot occur at the same time e.g. you can't roll a 2 and 5 on a single die at once.
- Independent events - Events where one outcome doesn't affect another outcome e.g. two separate coin tosses.
- Complementary events - Events where the occurrence of one event means the other cannot occur. For example, rolling an even number vs. an odd number on a die.
- Theoretical probability - Calculate the probability using all possible outcomes: P(A)=Number of Favorable Outcomes / Total Number of Possible Outcomes.
- Experimental probability - Probability that's based on the outcomes of an actual experiment: P(A) = Number of Times Event A Occurs / Total Number of Trials.
- Subjective probability - Probability that's based on personal judgment or experience instead of calculations.
- Axiomatic probability - Probability that's based on a set of axioms or rules, particularly those created by Andrey Kolmogorov.
- Conditional probability - How likely an event will happen given that another event has already occurred: P(A|B) = P(A intersection B) / P(B).
- Bayes' theorem - This formula lets you update the probability of an event using new evidence: P(A|B) = P(B|A) . P(A) / P(B).
