Basic Probability

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Basic Probability

Probability is a number between 0 (impossible) and 1 (certain).
Probability scale: 0 = impossible, 0.5 = even chance, 1 = certain.
P(A) = (number of outcomes in event A) / (total number of equally likely outcomes).
All probabilities sum to 1.
Complement: P(not A) = 1 – P(A).
Mutually exclusive events cannot happen together; P(A or B) = P(A) + P(B).
Probabilities can be fractions, decimals, or percentages.

Probability scale

Possibility (Sample Space) Diagrams

A sample space lists all possible outcomes of an experiment.
For two events (e.g., rolling two dice), use a grid (possibility diagram).
Each cell in the grid represents one equally likely outcome.
Probability = (number of desired outcomes) / (total outcomes in sample space).
Counting method works only if all outcomes are equally likely.
For three or more events, list outcomes systematically (e.g., HHH, HHT, ...).

Sample space for two dice

Relative Frequency

Relative frequency = (number of successful trials) / (total number of trials).
It estimates probability when theoretical probability is unknown.
More trials give a more accurate estimate (law of large numbers).
Compare relative frequency to theoretical probability to test fairness/bias.
Trials must be independent and random (e.g., replace items).

Expected Frequency

Expected frequency = probability × number of trials.
It predicts how many times an outcome will occur in a given number of trials.
Example: P(red) = 0.2, 500 trials → expected reds = 0.2 × 500 = 100.
If relative frequency is known, use it as the probability in the formula.
The best estimate comes from the experiment with the most trials.

Worked Examples

Example 1: Bag with 6 blue, 4 red, 5 yellow. P(yellow) = 5/15 = 1/3. Expected in 300 draws: 300 × 1/3 = 100.
Example 2: Biased coin flipped 40 times → 10 heads. Relative frequency = 10/40 = 0.25. Expected heads in 100 flips: 0.25 × 100 = 25.
Example 3: Two dice rolled. P(sum > 5 and odd) = 12/36 = 1/3. Given that, P(one die shows 2) = 2/12 = 1/6.

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Practice questions

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1.The probability that a sweet made in a factory is the wrong shape is 0.0028. One day, the factory makes 25 000 sweets. Calculate the number of sweets that are expected to be the wrong shape.
Easy
- A70
- B700
- C7
- D0.7
2.The probability that Kim wins a game is 0.72. In one year Kim will play 225 games. Work out an estimate of the number of games Kim will win.
Easy
- A162
- B1620
- C16.2
- D324
3.Sushila has a bag which contains 10 red balls and 8 blue balls. Sushila takes one ball at random from her bag. Find the probability that she takes a red ball.
Easy
- A10/18
- B8/18
- C10/8
- D1/2
4.The time taken for each of 120 students to complete a cooking challenge is shown in the table. Time (t minutes): 20<t≤25 (44), 25<t≤30 (32), 30<t≤35 (28), 35<t≤40 (12), 40<t≤45 (4). A student is chosen at random. Find the probability that this student takes more than 40 minutes.
Easy
- A4/120
- B12/120
- C28/120
- D44/120
5.The frequency table shows information about the time, m minutes, that each of 160 people spend in a library. Time (m minutes): 0<m≤10 (3), 10<m≤40 (39), 40<m≤60 (43), 60<m≤90 (55), 90<m≤100 (11), 100<m≤120 (9). Find the probability that one of these people, chosen at random, spends more than 100 minutes in the library.
Easy
- A9/160
- B11/160
- C20/160
- D55/160
6.On any given day the probability that it is sunny is 2/5. In a period of 90 days, on how many days is it expected to be sunny?
Easy
- A36
- B54
- C45
- D18
7.Sofia has a bag containing 8 blue beads and 7 red beads only. She takes one bead out of the bag at random and replaces it. She does this 90 times. Find the number of times she expects to take a red bead.
Medium
- A42
- B48
- C45
- D35
8.One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5. There are 36 male teachers at the school. Work out the total number of teachers at the school.
Medium
- A90
- B60
- C54
- D72

More questions

9.Rhiana plays a game. The probability that she will lose the game is 0.32. The probability that she will draw the game is 0.05. Rhiana is going to play the game 200 times. Work out an estimate for the number of times Rhiana will win the game.
Medium
- A126
- B64
- C10
- D136
10.The table shows the probabilities that a biased dice will land on 2, on 3, on 4, on 5 and on 6. Number on dice: 1 (0.17), 2 (0.18), 3 (0.09), 4 (0.15), 5 (0.1), 6 (missing). Neymar rolls the biased dice 200 times. Work out an estimate for the total number of times the dice will land on 1 or on 3.
Medium
- A52
- B34
- C18
- D68
11.A bag contains 7 red discs, 5 green discs and 2 pink discs. Helen takes one disc at random, records the colour and replaces it in the bag. She does this 140 times. Find how many times she expects to take a green disc.
Hard
- A50
- B35
- C70
- D20
12.There are only blue counters, green counters, red counters and yellow counters in a bag. Olga is going to take at random a counter from the bag. The table shows the probability that Olga will take a blue counter (0.4) and the probability that she will take a yellow counter (0.15). The number of red counters in the bag is 4 times the number of green counters in the bag. Complete the table to find the probability of taking a red counter.
Hard
- A0.36
- B0.09
- C0.24
- D0.45
13.There are some counters in a bag. The counters are red or white or blue or yellow. Bob is going to take at random a counter from the bag. The table shows each of the probabilities that the counter will be blue (0.25) or will be yellow (0.25). There are 18 blue counters in the bag. The probability that the counter Bob takes will be red is twice the probability that the counter will be white. Work out the number of red counters in the bag.
Hard
- A24
- B12
- C18
- D36
14.A marble is going to be taken at random from a box of marbles. The probability that the marble will be silver is 0.5. There must be an even number of marbles in the box. Explain why.
Hard
- ABecause half of the marbles are silver, so total must be even.
- BBecause 0.5 is a half, so the number of silver marbles is half the total, which must be an integer, so total is even.
- CBecause probability 0.5 means equal numbers of silver and non-silver, so total is twice the number of silver, which is even.
- DBecause the probability is 0.5, so the total number of marbles is a multiple of 2.
15.Four friends each throw a biased coin a number of times. The table shows the number of heads and tails: Ben (heads 34, tails 8), Helen (66,12), Paul (80,40), Sharif (120,40). The coin is to be thrown one more time. Which of the four friends' results will give the best estimate for the probability that the coin will land heads? Justify your answer.
Hard
- ABen, because he has the highest proportion of heads.
- BHelen, because she has the most heads.
- CPaul, because his results are balanced.
- DSharif, because he has the most trials.
16.A box contains 3 blue pens, 4 red pens and 8 green pens only. A pen is chosen at random from the box. Find the probability that this pen is green.
Easy
- A8/15
- B4/15
- C3/15
- D1/3
17.Samira has a bag containing 20 sweets. In the bag, there are 5 red, 6 green and 9 yellow sweets. Samira chooses one sweet at random from the bag. Write down the probability that she chooses a yellow sweet.
Easy
- A9/20
- B5/20
- C6/20
- D11/20
18.The probability that a biased dice will land on a five is 0.3. Megan is going to roll the dice 400 times. Work out an estimate for the number of times the dice will land on a five.
Easy
- A120
- B300
- C40
- D12
19.Jane has a packet of seeds. The probability that a seed will grow is 0.75. What is the probability that a seed will not grow?
Easy
- A0.25
- B0.75
- C0.5
- D0
20.The heights, in metres, of the 120 boys in an athletics club are recorded. The table shows: Height (h metres) 1.3<h≤1.4 (7), 1.4<h≤1.5 (18), 1.5<h≤1.6 (30), 1.6<h≤1.7 (24), 1.7<h≤1.8 (27), 1.8<h≤1.9 (14). One boy is chosen at random. Find the probability that this boy has a height greater than 1.8 m.
Easy
- A14/120
- B27/120
- C14/106
- D7/60
21.The test scores of 14 students are: 21, 21, 23, 26, 25, 21, 22, 20, 21, 23, 23, 27, 24, 21. A student is chosen at random. Find the probability that this student has a test score of more than 24.
Easy
- A3/14
- B4/14
- C5/14
- D2/14
22.A bag contains blue, red, yellow and green balls only. The table shows probabilities: Blue 0.15, Red 0.2, Yellow 0.43, Green ?. Complete the table. What is the probability of green?
Medium
- A0.22
- B0.25
- C0.78
- D0.12
23.There are 6 blue, 4 red and 5 yellow counters in a bag. One counter is drawn at random and replaced. Find the probability that a counter drawn is yellow.
Medium
- A5/15
- B4/15
- C6/15
- D1/3
24.If you flip a biased coin 40 times and get 10 heads, how many heads would you expect when flipping 100 times?
Medium
- A25
- B10
- C40
- D50
25.Two fair six-sided dice are rolled. Find the probability that the sum of the numbers showing is an odd number greater than 5.
Medium
- A1/3
- B5/18
- C7/36
- D1/6
26.A bag contains 7 red discs, 5 green discs and 2 pink discs. Helen takes one disc at random, records the colour and replaces it. She does this 140 times. Find how many times she expects to take a green disc.
Hard
- A50
- B35
- C70
- D25
27.There are only blue counters, green counters, red counters and yellow counters in a bag. The probability of blue is 0.4, yellow is 0.15. The number of red counters is 4 times the number of green counters. Complete the table: find the probability of red.
Hard
- A0.36
- B0.09
- C0.45
- D0.18
28.There are some counters in a bag: red, white, blue, yellow. The probabilities: blue 0.25, yellow 0.25. There are 18 blue counters. The probability that the counter is red is twice the probability that it is white. Work out the number of red counters.
Hard
- A12
- B6
- C18
- D24
29.Four friends each throw a biased coin a number of times. Results: Ben (heads 34, tails 8), Helen (66,12), Paul (80,40), Sharif (120,40). The coin is to be thrown one more time. Which of the four friends' results will give the best estimate for the probability that the coin will land heads? Justify your answer.
Hard
- ASharif, because he has the most trials.
- BBen, because his proportion of heads is highest.
- CHelen, because her ratio is closest to 5:1.
- DPaul, because his total is 120.
30.A bag contains 7 red discs, 5 green discs and 2 pink discs. Helen takes one disc at random, records the colour and replaces it in the bag. She does this 140 times. How many times does she expect to take a green disc?
Hard
- A50
- B40
- C30
- D20
31.There are only blue, green, red and yellow counters in a bag. The probability of blue is 0.4 and the probability of yellow is 0.15. The number of red counters is 4 times the number of green counters. Complete the table. What is the probability of green?
Medium
- A0.09
- B0.1
- C0.12
- D0.15
32.There are some counters in a bag: red, white, blue, yellow. The probability of blue is 0.25 and the probability of yellow is 0.25. There are 18 blue counters. The probability that the counter is red is twice the probability that it is white. Work out the number of red counters.
Hard
- A24
- B18
- C12
- D6
33.Four friends each throw a biased coin a number of times. The table shows the number of heads and tails: Ben (heads 34, tails 8), Helen (66,12), Paul (80,40), Sharif (120,40). The coin is to be thrown one more time. Which friend's results give the best estimate for the probability of heads?
Medium
- ABen
- BHelen
- CPaul
- DSharif
34.Four friends each throw a biased coin: Ben (heads 34, tails 8), Helen (66,12), Paul (80,40), Sharif (120,40). Paul says, 'With this coin you are twice as likely to get heads as to get tails.' Is Paul correct?
Medium
- AYes, because his own results show 80 heads and 40 tails
- BNo, because the overall relative frequency of heads is 300/400 = 0.75, not 2/3
- CNo, because the probability of heads is not exactly twice tails
- DYes, because 80/40 = 2
35.Four friends each throw a biased coin: Ben (heads 34, tails 8), Helen (66,12), Paul (80,40), Sharif (120,40). Use all results to estimate the probability that the coin lands heads both times when thrown twice.
Hard
- A0.5625
- B0.75
- C0.5
36.A box contains 3 blue pens, 4 red pens and 8 green pens only. A pen is chosen at random. Find the probability that it is green.
Easy
- A8/15
- B8/12
- C8/7
- D4/15
37.Samira has a bag containing 20 sweets: 5 red, 6 green, 9 yellow. She chooses one sweet at random. Write down the probability that she chooses a yellow sweet.
Easy
- A9/20
- B6/20
- C5/20
- D11/20

Revision notes

## Basic Probability ### Probability Scale - 0 = impossible, 1 = certain. - Can be fraction, decimal, or percentage. ### Key Definitions - **Experiment**: repeated activity; each repeat is a **trial**. - **Outcome**: possible result of a trial. - **Event**: an outcome or set of outcomes (e.g., rolling an even number). - **Sample space**: all possible outcomes. - **Fair**: all outcomes equally likely; **biased**: not equally likely. ### Calculating Probability If all outcomes equally likely: \[ P(A) = \frac{\text{number of outcomes for event }A}{\text{total number of outcomes}} \] ### Missing Probabilities - All probabilities sum to 1. - **Complement**: \( P(\text{not }A) = 1 - P(A) \). ### Mutually Exclusive Events - Cannot happen at the same time. - \( P(A \text{ or } B) = P(A) + P(B) \). ### Possibility (Sample Space) Diagrams - Grid for two sets of outcomes (e.g., two dice). - Count desired outcomes; divide by total. ### Relative Frequency - Estimate from experiment: \( \frac{\text{number of successful outcomes}}{\text{total number of trials}} \). - More trials → more accurate. - Used when theoretical probability unknown. ### Expected Frequency - \( \text{Expected frequency} = \text{probability} \times \text{number of trials} \). - Also: \( \text{relative frequency} \times \text{number of trials} \). ### Tips - Use fractions unless told otherwise. - For best estimate, choose experiment with most trials. - Replace items in bag for independent trials.

Probability: A measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
Sample space: The set of all possible outcomes of an experiment.
Event: An outcome or a collection of outcomes.
Mutually exclusive: Events that cannot happen at the same time.
Relative frequency: An estimate of probability based on experimental data: number of successful outcomes divided by total number of trials.
Expected frequency: The number of times an outcome is predicted to occur: probability multiplied by number of trials.
Complement: The event that an event does not happen; P(not A) = 1 - P(A).
Possibility diagram: A grid showing all possible outcomes of two events (e.g., rolling two dice).
Fair: When all outcomes have an equal chance of occurring.
Biased: When outcomes do not have equal chances; the probability is not uniform.

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