1.
A fair six-sided dice is rolled. Find P(even number).
[1 mark]
- 2, 4, 6
- Final answer: 1/2
JD SCIENCE
Probability
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Two fair coins are flipped. Find P(two heads).
[2 marks]
- HH, HT, TH, TT
- Final answer: 1/4
3.
A bag contains 8 red and 10 blue counters. One counter is taken at random. Find P(red).
[2 marks]
- Final answer: 8/18
4.
A spinner has sections A, A, B, C. It is fair.
(a) [1 mark]
- 1/2
(b) [2 marks]
- (1/2)×(1/4) = 1/8
5.
P(rain) = 0.3. Find P(not rain).
[1 mark]
- 1 − 0.3
- Final answer: 0.7
6.
Events A and B are mutually exclusive. P(A) = 0.2, P(B) = 0.5. Find P(A or B).
[2 marks]
- 0.7
- Add because they cannot both happen
7.
Events C and D are independent. P(C) = 0.4, P(D) = 0.5. Find P(C and D).
[2 marks]
- 0.2
8.
A bag has 3 red and 5 green sweets. Two sweets are taken without replacement. Find P(both red).
[3 marks]
- Tree diagram
- Final answer: 3/8 × 2/7 = 6/56 = 3/28
9.
Using the same bag, find P(one of each colour) when two sweets are taken without replacement.
[4 marks]
- Two orders
- Final answer: 3/8×5/7 + 5/8×3/7 = 30/56 = 15/28
10.
Draw a complete tree diagram for two flips of a biased coin with P(H) = 0.6. Label all eight probabilities on the branches and ends.
[4 marks]
- Branches 0.6 and 0.4 twice
- HH 0.36, HT 0.24, TH 0.24, TT 0.16
11.
A two-way table: 20 students, 12 study French, 9 study German, 5 study both. How many study neither?
[3 marks]
- Inclusion-exclusion: 12+9−5 = 16, 20−16 = 4
- Final answer: 4
12.
From that table, a student is chosen at random. Find P(studies French | studies German).
[3 marks]
- Conditional: both / German
- Final answer: 5/9
13.
Write the formula for P(A|B) and explain it in words.
[2 marks]
- P(A and B) / P(B)
- Restrict the sample space to B
14.
The probability of winning a game is 0.15. The game is played 200 times. Estimate the number of wins.
[2 marks]
- Expected frequency np
- Final answer: 30
15.
Relative frequency of a 6 after 50 rolls is 0.1. After 500 rolls it is 0.16. Which is the better estimate of P(6) on a fair dice, and why?
[3 marks]
- 500 rolls
- More trials, relative frequency closer to the true probability
16.
A letter is chosen from MISSISSIPPI. Find P(S).
[2 marks]
- 4 S letters, 11 total
- Final answer: 4/11
17.
Two events: P(A) = 0.3, P(B) = 0.4, P(A and B) = 0.12. Are A and B independent? Show working.
[3 marks]
- Final answer: Yes, because 0.3×0.4 = 0.12
18.
P(A) = 0.6, P(B) = 0.5, P(A and B) = 0.2. Find P(A or B).
[2 marks]
- 0.6+0.5−0.2
- Final answer: 0.9
19.
What is a sample space? List the sample space when a coin is flipped and a dice is rolled.
[3 marks]
- All possible outcomes
- H1–H6 and T1–T6, 12 outcomes
20.
A fair spinner 1–5 is spun twice. Find P(sum is 10).
[2 marks]
- Only 5 then 5
- Final answer: 1/25
21.
Find P(sum is 6) for two fair dice.
[3 marks]
- (1,5)(2,4)(3,3)(4,2)(5,1)
- Final answer: 5/36
22.
Explain the difference between independent events and mutually exclusive events. Give an example of each.
[4 marks]
- Independent: one does not affect the other, e.g. two coin flips
- Mutually exclusive: cannot happen together, e.g. rolling a 2 and a 5 on one roll
23.
Owain picks a number from 1 to 20. Find P(prime).
[2 marks]
- 2,3,5,7,11,13,17,19
- Final answer: 8/20 = 2/5
24.
A Venn diagram has P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.15.
(a) [2 marks]
- 0.35
(b) [2 marks]
- 0.25
25.
Without replacement: 10 tickets numbered 1–10. Two tickets are drawn. Find P(both even).
[3 marks]
- Final answer: 5/10 × 4/9 = 2/9
26.
Why must probabilities on a set of mutually exclusive exhaustive outcomes add up to 1?
[2 marks]
- One of them must happen
- The sample space is complete
27.
A biased dice has P(6) = 0.3. It is rolled twice. Find P(at least one 6).
[3 marks]
- Complement
- Final answer: 1 − 0.7² = 0.51
28.
Expected winnings: a game costs £2. P(win £10) = 0.1, otherwise win nothing. Is the game fair? Show working.
[3 marks]
- Final answer: Expected gain = 0.1×10 − 2 = −1, not fair
29.
Describe how to use a set of random numbers from 00 to 99 to simulate P(success) = 0.27.
[2 marks]
- Assign 00–26 as success
- 27–99 as failure
30.
A WJEC tree-diagram question is often 4 marks. State what must be labelled on the branches and on the ends.
[2 marks]
- Branch probabilities
- Outcome probabilities multiplied along the path
31.
Three fair coins. Find P(exactly two heads).
[3 marks]
- HHT, HTH, THH
- Final answer: 3/8
32.
The probability of an event is 0. Explain what this means. The probability is 1. Explain what this means.
[2 marks]
- Impossible
- Certain