Probability
Grade 2-4Basic Probability & Sample Spaces
Every probability question comes back to the same idea, the number of outcomes you want divided by the total number of possible outcomes, and the skill is being organised enough to count both correctly. This lesson covers finding the probability of a single event, listing outcomes of combined events, using sample-space diagrams for two combined events, and finding a missing probability.
What you need to know
- Probability is the number of favourable outcomes divided by the total number of equally likely outcomes.
- Every probability is between 0 (impossible) and 1 (certain).
- The probabilities of all possible outcomes add up to 1.
- P(not A) equals 1 minus P(A).
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the probability of a single event
Probability is calculated as the number of favourable outcomes divided by the total number of possible outcomes, written as a fraction and simplified fully.
A bag contains 5 red, 3 blue and 2 green counters, 10 counters in total. The probability of picking red is 5/10, which simplifies to 1/2.
Listing outcomes of combined events
When two events happen together, such as flipping a coin and spinning a spinner, list every possible combination systematically, working through each outcome of the first event with each outcome of the second, to avoid missing or repeating any.
Flipping two coins gives the outcomes HH, HT, TH and TT, four outcomes in total, found by pairing each result of the first coin with each result of the second.
Using a sample-space diagram
A sample-space diagram is a grid showing every possible outcome of two combined events, such as two dice rolled together. Count how many cells in the grid match the outcome you want, then divide by the total number of cells.
Rolling two 4-sided dice numbered 1 to 4 and adding the scores gives 16 total outcomes. A total of 5 appears in 4 of these cells, giving a probability of 4/16, which simplifies to 1/4.
Finding a missing probability
The probabilities of all the possible outcomes for an event always add up to 1. To find a missing probability, add up the probabilities already given and subtract the total from 1.
A spinner lands on red, blue, green or yellow, with P(red) = 0.3, P(blue) = 0.25 and P(green) = 0.2. Since all four probabilities add to 1, P(yellow) = 1 - 0.3 - 0.25 - 0.2 = 0.25.
Worked examples
Three exam-style questions, fully solved
A bag contains 5 red counters, 3 blue counters and 2 green counters. A counter is picked at random. Find the probability that it is red.
Easy- 1.Find the total number of counters: 5 + 3 + 2 = 10
- 2.Write the probability as favourable outcomes over total outcomes: 5/10
- 3.Simplify the fraction: 5/10 = 1/2
Answer: 1/2
A spinner can land on red, blue, green or yellow. P(red) = 0.3, P(blue) = 0.25, P(green) = 0.2. Work out P(yellow).
Medium- 1.Recall that all the probabilities must add up to 1
- 2.Add the known probabilities: 0.3 + 0.25 + 0.2 = 0.75
- 3.Subtract from 1: 1 - 0.75
Answer: 0.25
Two fair 4-sided dice, each numbered 1 to 4, are rolled and their scores are added together. Use a sample-space diagram to find the probability that the total is 5.
Hard- 1.Draw a 4 by 4 grid showing every possible total, giving 16 outcomes altogether
- 2.Count how many cells show a total of 5: 4 cells
- 3.Write the probability as a fraction and simplify: 4/16 = 1/4
Answer: 1/4
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A fair six-sided dice is rolled. What is the probability of getting a 4?
A fair six-sided dice is rolled. What is the probability of getting an even number?
A bag contains 5 red counters and 3 blue counters. One is taken at random. What is the probability it is blue?
A letter is picked at random from the word BANANA. What is the probability it is an A?
The probability that it rains tomorrow is 0.3. What is the probability that it does not rain?
A spinner has ten equal sections numbered 1 to 10. What is the probability of spinning a multiple of 3?
Avoid these
The mistakes students make most often
Writing a probability as a raw count, like "5 counters", instead of as a fraction of the total number of outcomes.
Forgetting to simplify the resulting fraction fully, leaving an answer like 5/10 instead of 1/2.
Missing an outcome when listing combined events, or accidentally listing the same outcome twice, instead of working through the list systematically.
In a sample-space diagram, only counting some of the cells that match the target outcome instead of checking the entire grid.
Forgetting that all the probabilities for a set of outcomes must add up to 1, and adding the given probabilities instead of subtracting them from 1.
FAQ
Questions students and parents ask
What to learn next
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