🎲 Probability

Grade 8 · Statistics & Probability · Cambridge Lower Secondary

Probability Scale

0 = impossible, 1 = certain

Calculating P(event)

P = favourable ÷ total

Complement Rule

P(not A) = 1 − P(A)

Mutually Exclusive

P(A or B) = P(A) + P(B)

Expected Frequency

Expected = P × n

1. The Probability Scale

Probability measures how likely an event is. It always lies between 0 and 1 (inclusive).

0
Impossible
0.25
Unlikely
0.5
Even chance
0.75
Likely
1
Certain
Examples:
P(rolling a 7 on a die) = 0 — impossible
P(rolling 1, 2, 3, 4, 5, or 6) = 1 — certain
P(flipping heads) = 0.5 — even chance
Probability can be written as a fraction, decimal, or percentage. 1/4 = 0.25 = 25%

2. Calculating Probability

P(event) = Number of favourable outcomes ÷ Total number of equally likely outcomes

This assumes all outcomes are equally likely (fair dice, well-shuffled cards, etc.)

A bag has 3 red, 5 blue, 2 green balls. One is chosen at random.
Total = 10. P(red) = 3/10 = 0.3. P(blue) = 5/10 = 1/2. P(green) = 2/10 = 1/5.
SituationP(event)
Roll a 4 on a fair die1/6
Draw a heart from 52 cards13/52 = 1/4
Pick a vowel from A,B,C,D,E2/5
Roll an even number3/6 = 1/2

3. The Complement Rule

The complement of event A is "not A" — everything that is NOT event A.

P(not A) = 1 − P(A)
P(rain tomorrow) = 0.35. Then P(no rain) = 1 − 0.35 = 0.65

This is useful when it's easier to find P(A) and then subtract from 1.

All outcomes form a complete set: P(A) + P(not A) = 1. Always.

4. Mutually Exclusive Events

Two events are mutually exclusive if they cannot both happen at the same time.

P(A or B) = P(A) + P(B)   [if mutually exclusive]
Rolling a die: P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 2/6 = 1/3
These are mutually exclusive because you can't roll both 2 and 5 simultaneously.

All mutually exclusive outcomes: If A, B, C are all possible outcomes and mutually exclusive:

P(A) + P(B) + P(C) = 1
P(red) = 0.3, P(blue) = 0.5, P(green) = ?
P(green) = 1 − 0.3 − 0.5 = 0.2
On a probability scale: if events are mutually exclusive and exhaustive, their probabilities add to 1.

5. Expected Frequency

If an experiment is repeated n times, the expected frequency of an event is:

Expected frequency = P(event) × n
A biased coin has P(heads) = 0.6. Flipped 200 times:
Expected heads = 0.6 × 200 = 120

Expected frequency is a prediction, not a guarantee. Actual results will vary.

If a die is rolled 300 times, expected number of 6s = (1/6) × 300 = 50.

Example 1 — Basic Probability

A bag contains 4 red, 6 blue, and 2 yellow counters. One is picked at random.

Total outcomes: 4 + 6 + 2 = 12
P(red) = 4/12 = 1/3
P(blue) = 6/12 = 1/2
P(yellow) = 2/12 = 1/6
Check: 1/3 + 1/2 + 1/6 = 2/6 + 3/6 + 1/6 = 6/6 = 1 ✓

Example 2 — Complement Rule

A spinner has sectors: P(red) = 0.4, P(blue) = 0.35. Find P(not blue) and P(green) if there are only three colours.

P(not blue) = 1 − P(blue) = 1 − 0.35 = 0.65
P(green) = 1 − P(red) − P(blue) = 1 − 0.4 − 0.35 = 0.25

Example 3 — Mutually Exclusive Events

P(A) = 1/4, P(B) = 1/6. Events A and B are mutually exclusive. Find P(A or B).

P(A or B) = P(A) + P(B) = 1/4 + 1/6 = 3/12 + 2/12 = 5/12

Example 4 — Expected Frequency

A spinner lands on Red with probability 0.3. It is spun 500 times.

Expected Red = 0.3 × 500 = 150
If P(Blue) = 0.45, Expected Blue = 0.45 × 500 = 225
P(Green) = 1 − 0.3 − 0.45 = 0.25, Expected Green = 0.25 × 500 = 125
Check: 150 + 225 + 125 = 500 ✓

🎲 Probability Calculator

🔄 Complement & Mutually Exclusive

Enter up to 4 event probabilities (must sum ≤ 1):

📊 Expected Frequency Simulator

Exercise 1 — Basic Probability (as decimals)

A bag has 5 red, 3 blue, 2 green balls. Give answers as decimals.

1. P(red)?

2. P(blue)?

3. P(green)?

4. P(not red)?

5. P(not green)?

A fair die (1–6):

6. P(rolling 3)?

7. P(rolling even)?

8. P(rolling > 4)?

9. P(rolling ≤ 2)?

10. P(rolling a prime: 2,3,5)?

Exercise 2 — Complement Rule

1. P(A) = 0.7. Find P(not A).

2. P(rain) = 0.45. Find P(no rain).

3. P(win) = 1/4. Find P(not win) as a decimal.

4. P(B) = 3/8. P(not B) as a decimal?

5. P(red) = 0.3, P(blue) = 0.5. P(green) if only 3 colours?

6. P(A) = 0.35, P(B) = 0.4. P(neither A nor B) if mutually exclusive and exhaustive with C?

7. P(not E) = 0.82. Find P(E).

8. P(not F) = 0.05. Find P(F).

9. P(A) = 2/5. P(not A) as a decimal?

10. P(X) = 0.125. Find P(not X).

Exercise 3 — Mutually Exclusive Events

All events are mutually exclusive.

1. P(A) = 1/3, P(B) = 1/4. P(A or B) as a decimal?

2. P(X) = 0.2, P(Y) = 0.35. P(X or Y)?

3. P(A) = 0.4, P(B) = 0.25. P(A or B)?

4. P(A) = 1/6, P(B) = 1/6, P(C) = 1/6. P(A or B or C)?

5. P(A) = 0.15, P(B) = 0.3, P(C) = 0.4. Find P(D) if A,B,C,D exhaustive.

6. P(A or B) = 0.7, P(A) = 0.3. Find P(B).

7. P(A or B) = 5/8, P(B) = 1/4. Find P(A) as a decimal.

8. 3 mutually exclusive events, all equal probability. Each P?

9. P(A) = 0.28, P(B) = 0.35, P(C) = 0.17. P(A or B or C)?

10. P(not A) = 0.55. P(A) in a 2-outcome system?

Exercise 4 — Expected Frequency

1. P = 0.5, n = 100. Expected frequency?

2. P = 1/6, n = 120. Expected frequency?

3. P = 0.3, n = 200. Expected frequency?

4. P = 0.25, n = 80. Expected frequency?

5. P = 3/5, n = 150. Expected frequency?

6. P = 0.04, n = 500. Expected frequency?

7. Expected frequency = 35, n = 140. Find P.

8. P = 0.6, expected = 90. Find n.

9. P = 1/4, n = 60. Expected frequency?

10. P = 0.35, n = 400. Expected frequency?

Exercise 5 — Mixed

1. P(A) = 0.45. P(not A)?

2. 8 equally likely outcomes. P(one specific outcome)?

3. P(A) = 2/7, P(B) = 3/7 (mutually exclusive). P(A or B)?

4. P = 0.7, n = 300. Expected frequency?

5. Die (1–6). P(rolling < 3)?

6. P(A) = 0.4, P(B) = 0.35, A and B mutually exclusive. P(neither)?

7. P = 3/4, n = 200. Expected frequency?

8. Expected frequency = 60, P = 0.4. Find n.

9. P(X) = 0.38. P(not X)?

10. 4 mutually exclusive equally likely outcomes. P(any one)?

🏋️ Practice — 20 Questions

1. P(A) = 0.65. P(not A)?

2. P = 1/3, n = 90. Expected?

3. P(A) = 0.4, P(B) = 0.3 (ME). P(A or B)?

4. Bag: 4R, 3B, 3G. P(red)?

5. P = 0.25, n = 180. Expected?

6. P(not B) = 0.72. P(B)?

7. P(A) = 3/8, P(B) = 1/8 (ME). P(A or B) as decimal?

8. Die 1–6. P(multiple of 3)?

9. Expected = 45, n = 300. P?

10. P(X) = 0.56, P(Y) = 0.24 (ME). P(X or Y)?

11. 5 equally likely outcomes. P(one)?

12. P = 0.8, n = 150. Expected?

13. P(A) = 0.3, P(B) = 0.45, P(C) = ? (exhaustive ME). P(C)?

14. P(A or B) = 0.65, P(A) = 0.4 (ME). P(B)?

15. P = 2/9, n = 270. Expected?

16. Bag: 7R, 3B. P(blue) as decimal?

17. P(not E) = 3/7. P(E) as decimal?

18. P = 0.45, n = 400. Expected?

19. Expected = 150, P = 0.6. n?

20. P(A) = 1/5. P(not A) as decimal?

🏆 Challenge — 8 Questions

1. P(A) = 3x, P(B) = 2x, P(C) = x. Events are mutually exclusive and exhaustive. Find P(A). (Enter as decimal)

2. A bag has red and blue balls only. P(red) = 0.35. If there are 20 balls, how many are blue?

3. A spinner is spun 250 times. P(green) = 0.28. Expected green? What is the probability of NOT landing on green in a single spin? (Enter expected frequency)

4. P(A) = 0.45, P(B) = 0.3, A and B are mutually exclusive. Find P(A or B).

5. A die is rolled 180 times. How many times would you expect a number less than 3?

6. P(A) = 2k+0.1, P(B) = k+0.2. A and B are mutually exclusive and exhaustive (only two outcomes). Find k. (Enter as decimal)

7. A bag has 3 red, 5 blue, 2 green. If one ball is taken out (blue) and NOT replaced, what is P(green) now? (Enter as decimal, 2 d.p.)

8. Expected frequency of event X is 84 when n = 350. Find P(not X). (Enter as decimal)