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Probability measures how likely an event is to happen, on a scale from 0 to 1.
A probability of 0 means the event is impossible; a probability of 1 means it is certain.
Probabilities can be written as fractions, decimals, or percentages (e.g., 1/2, 0.5, 50%).
The closer a probability is to 1, the more likely the event; the closer to 0, the less likely.
Use words like impossible, unlikely, even chance, likely, and certain to describe probabilities on the scale.
Probability scale
Theoretical Probability
For equally likely outcomes, the theoretical probability of an event is: P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}.
Example: rolling a fair 6-sided die, the probability of getting a 4 is \frac{1}{6}, because there is 1 favourable outcome out of 6 equally likely outcomes.
The probability of an event is always between 0 and 1 inclusive, so it can never be negative or greater than 1.
When all outcomes are equally likely, each outcome has the same probability. For example, each face of a fair die has probability \frac{1}{6}.
If outcomes are not equally likely, you cannot use the simple fraction method; you need other information (e.g., from experiments).
Equally likely die outcomes
Probability of an Event Not Happening
The probabilities of all possible outcomes of a situation add up to 1.
The probability that an event does not happen is: P(\text{not } A) = 1 - P(A).
Using the complement is often quicker than listing all other outcomes.
Example: if the probability of rain tomorrow is 0.3, then the probability of no rain is 1 - 0.3 = 0.7.
This works because the event and its complement are mutually exclusive and cover all possibilities.
An event and its complement
Mutually Exclusive and Exhaustive Events
Events are mutually exclusive if they cannot happen at the same time. For example, getting a head and getting a tail on one coin toss.
Events are exhaustive if they cover all possible outcomes. For example, getting a head or a tail covers all outcomes of a coin toss.
For a set of mutually exclusive and exhaustive events, their probabilities sum to 1.
If you know the probabilities of all but one event in such a set, you can find the missing probability by subtracting the known ones from 1.
Example: if the probabilities of a spinner landing on red, blue, and green are 0.2, 0.5, and unknown, then green is 1 - 0.2 - 0.5 = 0.3.
Experimental Probability and Relative Frequency
Experimental probability (or relative frequency) is found from carrying out an experiment: \text{relative frequency} = \frac{\text{frequency of the outcome}}{\text{total number of trials}}.
Example: if a coin is tossed 100 times and lands on heads 47 times, the relative frequency of heads is \frac{47}{100} = 0.47.
Relative frequency is an estimate of the true probability; as the number of trials increases, it tends to get closer to the theoretical probability.
Experimental probability can be used when theoretical probabilities are unknown or outcomes are not equally likely.
Do not confuse frequency (the count of how many times something happened) with probability (the chance of it happening).
Experimental spinner results
Recording Outcomes: Frequency Tables and Frequency Trees
A frequency table records how often each outcome occurs in an experiment.
A frequency tree shows the frequencies of outcomes at each stage of a multi-stage experiment, branching out from a starting point.
Frequency trees help organise data and calculate probabilities by dividing each frequency by the total number of trials.
Example: in a survey of 50 students, if 30 are girls and 20 are boys, a frequency tree can show these counts and further splits (e.g., girls who wear glasses).
Always check that the frequencies in a table or tree add up to the total number of trials.
Randomness, Fairness, and Equally Likely Outcomes
Random means the outcome is governed by chance. Outcomes need not be equally likely, and independence is a separate assumption.
Fair means all outcomes are equally likely, such as a fair coin or a fair die.
Assuming outcomes are equally likely when they are not is a common error. For example, the outcomes 'win' and 'lose' are not equally likely in most games.
In a fair situation, the theoretical probability can be calculated using the fraction method.
A biased object has unequal outcome probabilities. These may be given by a model or estimated through experiments; experimental data are not the only possible source.
The Gambler's Fallacy and Independence
The gambler's fallacy is the mistaken belief that past results influence the next independent trial.
For independent events, the outcome of one trial does not affect the outcome of the next. For example, if a fair coin lands on heads 5 times in a row, the probability of heads on the next toss is still \frac{1}{2}.
Each trial of a fair experiment has the same probability, regardless of previous results.
Do not expect that a 'losing streak' makes a win more likely in a fair game.
Independence must be stated or justified. In a standard well-mixed draw-with-replacement model, the probabilities reset before each draw.
Common Misconceptions to Avoid
Use a fraction, decimal or percentage for a GCSE probability answer. Odds such as 3 : 5 compare favourable and unfavourable outcomes, giving probability 3/8.
Do not give a probability greater than 1 or negative; that is impossible.
Do not assume all outcomes are equally likely unless stated (e.g., 'fair', 'biased').
Do not confuse the probability of an event with its frequency; frequency is a count, probability is a proportion.
Always consider using the complement (1 minus probability) when it is quicker than listing all other outcomes.
Typical Exam Questions
Find the probability of a simple event, such as picking a red card from a standard deck: \frac{26}{52} = \frac{1}{2}.
Complete a probability table where one value is missing by using the fact that probabilities sum to 1.
Find the probability that an event does not occur using 1 - P(A).
Complete and interpret a frequency tree to find probabilities of combined outcomes.
Explain why two outcomes are not equally likely, e.g., a biased spinner has unequal sections.
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Xem trước miễn phí — 8 trên 63 câu hỏi. Đăng ký để xem tất cả.
1.On the probability scale, what value represents an event that is certain to happen?
Easy
A0
B0.5
C1
D100
2.Which of the following is a valid probability for an event?
Easy
A-0.2
B1.5
C0.7
D7/5
3.The probability of an event that is impossible is 0.
Easy
True or false?
4.A bag contains 3 red, 2 blue and 5 green marbles. One marble is taken at random. What is the probability that it is blue?
Easy
A2/10
B2/8
C1/4
D2/3
5.Which of the following statements are true? (Select all that apply)
Medium
AThe probabilities of all possible outcomes of a trial add up to 1.
BIf an event has probability 0.3, then the probability of it not happening is 0.7.
CIf an independently tossed fair coin has landed heads 5 times, tails is more likely on the next toss.
DA probability can be written as a fraction, decimal or percentage.
6.Match each probability to its description on the probability scale.
Medium
0
0.5
1
Impossible
Even chance
Certain
7.Order these events from least likely to most likely: A: rolling a 6 on a fair dice, B: getting a head when flipping a fair coin, C: rolling a number less than 5 on a fair dice.
Medium
A
B
C
8.A biased spinner has three colours: red, blue and green. P(red) = 0.2 and P(blue) = 0.5. What is P(green)?
Hard
A0.3
B0.7
C0.2
D0.5
Thêm câu hỏi
9.If a fair coin is tossed 10 times and lands heads every time, the probability of tails on the next toss is greater than 0.5.
Medium
True or false?
10.A frequency tree shows that out of 100 students, 60 are girls and 40 are boys. Among the girls, 30 wear glasses. What is the probability that a randomly chosen student is a girl who wears glasses?
Hard
A0.3
B0.5
C0.6
D0.75
11.Which of the following statements about probabilities are correct? (Select all that apply)
Medium
AThe probability of an event can be greater than 1.
BThe sum of probabilities of all possible outcomes is 1.
CIf an event has probability 0, it is impossible.
DThe probability of an event can be written as a percentage.
12.Order these probabilities from smallest to largest: 0.3, 1/4, 40%, 0.2.
Medium
0.3
1/4
40%
0.2
13.Match each event to its probability on a fair six-sided dice.
Easy
Rolling an even number
Rolling a number less than 3
Rolling a 7
1/2
1/3
0
14.The probability of an event happening is 0.8, so the probability of it not happening is 0.2.
Medium
True or false?
15.Which word describes an event with probability 0?
Easy
ACertain
BLikely
CUnlikely
DImpossible
16.Which of the following are correct ways to write the probability of an event? (Select all that apply)
Medium
A0.5
B50%
C1/2
D5:10
17.Order these events from most likely to least likely: A: getting a head when flipping a fair coin, B: rolling a 6 on a fair dice, C: rolling a number less than 5 on a fair dice.
Medium
A
B
C
18.Order these events from least likely to most likely: A: rolling a 6 on a fair dice, B: getting a head when flipping a fair coin, C: rolling a number less than 5 on a fair dice.
Medium
A: rolling a 6 on a fair dice
B: getting a head when flipping a fair coin
C: rolling a number less than 5 on a fair dice
19.The probability of an event that is certain to happen is 1.
Easy
True or false?
20.A frequency tree shows that 80 students were surveyed. 30 said they liked tea, and of those, 20 also liked coffee. Among the 50 who did not like tea, 10 liked coffee. What is the probability that a randomly chosen student likes coffee?
Hard
A30/80
B20/80
C10/80
D50/80
21.Which of the following are true about the probability scale? (Select all that apply)
Medium
AA probability of 0.5 means the event is certain.
BA probability of 0 means the event is impossible.
CProbabilities can be written as fractions, decimals, or percentages.
DThe probability scale goes from 0 to 1.
EA probability of 1.2 is possible.
22.Match each probability to its equivalent percentage or fraction.
Medium
0.25
0.75
0.1
25%
3/4
10%
23.Order these events from most likely to least likely: A: heads on a fair coin, B: a six on a fair six-sided die, C: a heart from a standard 52-card pack.
Medium
A: heads on a fair coin
B: a six on a fair six-sided die
C: a heart from a standard 52-card pack
24.A bag contains 4 red, 3 blue, and 5 green counters. One counter is taken at random. What is the probability that it is not blue? Give your answer as a fraction.
Medium
A2/3
B3/12
C9/12
D5/12
25.The probability of an event is 0.6. The probability of the event not happening is 0.4.
Easy
True or false?
26.Which of the following are valid probabilities? (Select all that apply)
Medium
A0.5
B3/2
C25%
D1.2
E0
27.Order these probabilities from smallest to largest: 0.7, 1/4, 60%, 0.2
Easy
0.7
1/4
60%
0.2
28.Among 60 students, 35 have a cat and 25 do not. Of the cat owners, 20 also have a dog; of the students without a cat, 5 have a dog. What is the probability that a randomly chosen student has a dog? Give the simplest fraction.
Medium
A5/12
B20/60
C5/60
D25/35
29.A bag contains 6 red, 4 blue, and 2 green counters. A counter is taken at random. What is the probability that it is not blue? Give your answer as a fraction in its simplest form.
Medium
A\frac{2}{3}
B\frac{1}{3}
C\frac{4}{6}
D\frac{3}{4}
30.A frequency tree shows that 120 students were asked whether they walk to school. 70 students walk, and of those, 40 are girls. Among the 50 students who do not walk, 20 are boys. What is the probability that a randomly chosen student is a girl who walks to school? Give your answer as a fraction in its simplest form.
Medium
A\frac{1}{3}
B\frac{7}{12}
C\frac{2}{5}
D\frac{4}{7}
31.A fair coin is tossed 5 times and lands on heads each time. The probability of getting tails on the next toss is greater than 0.5.
Easy
True or false?
32.Which of the following are correct statements about probability? (Select all that apply)
Medium
AThe probability of an event can be 1.5.
BThe probability of an event that is certain is 1.
CThe probability of an event that is impossible is 0.
DThe probability of an event can be written as a percentage greater than 100%.
EThe sum of probabilities of all possible outcomes is 1.
33.Order these events from least likely to most likely. A: Rolling a 5 on a fair six-sided dice. B: Getting a head when flipping a fair coin. C: Rolling a number less than 5 on a fair six-sided dice.
Medium
A
B
C
34.A spinner was spun 100 times: red 20, blue 30, green 15 and yellow 35. Estimate P(red or green) as a fraction in its simplest form.
Medium
A\frac{7}{20}
B3/20
C\frac{1}{2}
D\frac{3}{10}
35.A fair spinner has four equal sectors: red, blue, green and yellow. Its theoretical probability of landing on yellow is 0.25.
Easy
True or false?
36.Which of the following is the correct range of values for a probability?
Easy
A0 to 1 inclusive
B0 to 1 exclusive
C-1 to 1 inclusive
D0 to 100 inclusive
37.Odds in favour of an event of 3 : 5 mean that its probability is 3/5.
Easy
True or false?
38.Which of the following is NOT a valid probability?
Easy
A0.5
B7/8
C1.2
D0
39.A fair six-sided dice is rolled. Which of the following events has a probability of 1/2?
Medium
AGetting an even number
BGetting a number greater than 4
CGetting a square number
DGetting a multiple of 3
40.Match each probability value to its equivalent percentage.
Easy
0.25
0.5
0.75
25%
50%
75%
41.Complete the sentence using the correct word from the list: certain, impossible, likely, unlikely.
Easy
An event with probability 0.9 is ____.
42.A spinner has 4 equal sectors: red, blue, green, yellow. What is the probability of landing on red or blue? Write your answer as a fraction.
Medium
43.The probability that it rains on a given day is 0.35. What is the probability that it does not rain?
Medium
44.Complete the probability table. The events A, B and C are mutually exclusive and exhaustive.
Medium
Event: A (0.2), B (0.5), C (____)
45.A bag contains only red and blue counters. The probability of picking a red counter is 0.3. If there are 20 counters in total, how many blue counters are there?
Medium
46.A probability experiment is repeated 200 times. A particular outcome occurs 50 times. What is the relative frequency of that outcome? Give your answer as a decimal.
Easy
Relative frequency = ____
47.A card is drawn at random from a standard pack of 52 playing cards. What is the probability of drawing a heart? Write your answer as a fraction.
Medium
48.A bag contains 5 red, 3 blue and 2 green marbles. One marble is taken at random. What is the probability that it is not blue? Give your answer as a decimal.
Medium
49.A frequency tree shows that 80 students were surveyed. 30 said they liked tea, and of those, 20 also liked coffee. Among the 50 who did not like tea, 10 liked coffee. Complete the frequency tree: How many students liked neither tea nor coffee?
Medium
Number who liked neither = ____
50.In a probability experiment, a spinner is spun 150 times. The spinner has 5 equal sections numbered 1 to 5. How many times would you expect it to land on an odd number?
Hard
51.A bag contains 3 red, 2 blue and 5 green marbles. One marble is taken at random. What is the probability that it is not blue? Give your answer as a decimal.
Medium
52.A bag contains 3 red, 2 blue and 5 green marbles. One marble is taken at random. What is the probability that it is blue? Give your answer as a decimal.
Medium
53.Explain why the outcomes 'heads' and 'tails' when tossing a fair coin are equally likely.
Medium
54.Explain why the outcomes 'win' and 'lose' are not necessarily equally likely in a football match.
Hard
55.Explain why the outcomes 'rain' and 'no rain' for a day in the UK are not necessarily equally likely.
Hard
56.Explain why the outcomes 'heads' and 'tails' when tossing a fair coin are equally likely, but the outcomes 'rain' and 'no rain' for a day in the UK are not necessarily equally likely.
Hard
57.Explain why the outcomes 'win' and 'lose' in a football match are not necessarily equally likely.
Medium
58.The probability that it rains on a given day is 0.35. What is the probability that it does not rain?
Medium
A-0.35
B2.65
C0.65
D1.65
59.A bag contains only red and blue counters. The probability of picking a red counter is 0.3. If there are 20 counters in total, how many blue counters are there?
Medium
A14
B16
C15
D13
60.A bag contains 5 red, 3 blue and 2 green marbles. One marble is taken at random. What is the probability that it is not blue? Give your answer as a decimal.
Medium
A1.7
B2.7
C-0.3
D0.7
61.In a probability experiment, a spinner is spun 150 times. The spinner has 5 equal sections numbered 1 to 5. How many times would you expect it to land on an odd number?
Hard
A90
B108
C99
D81
62.A bag contains 3 red, 2 blue and 5 green marbles. One marble is taken at random. What is the probability that it is not blue? Give your answer as a decimal.
Medium
A1.8
B2.8
C0.8
D-0.2
63.A bag contains 3 red, 2 blue and 5 green marbles. One marble is taken at random. What is the probability that it is blue? Give your answer as a fraction.
Medium
A\frac{3}{10}
B\frac{1}{2}
C\frac{1}{10}
D\frac{1}{5}
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