Probability Diagrams

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Two-Way Tables

Two-way tables compare two characteristics (e.g., year group and language).
Add row and column totals, including an overall total in the bottom-right corner.
For a random selection from the whole group, probability = (number in category) / (overall total).
For a random selection from a specific category, denominator is that category's total.
Check that row and column totals sum to the overall total to avoid errors.

Two-way table: language choice by year group

Probabilities from Venn Diagrams

If the Venn diagram shows frequencies, probability = (sum of frequencies in region) / (total frequency).
If the diagram shows individual elements, probability = (number of elements in region) / (total elements).
For conditional probability (e.g., P(B given A)), use only elements in A as the denominator.
When filling numbers, some given totals may need to be split between overlapping regions (e.g., '10 have a cat, 6 have both cat and dog' means 4 have cat only).

Sets A and B

Probability Tree Diagrams

Tree diagrams show outcomes of repeated experiments with two or more stages.
Write probabilities on each branch; P(not A) = 1 - P(A); probabilities on each pair of branches sum to 1.
Multiply along branches to find P(first outcome and second outcome).
Add probabilities of separate cases for 'or' events (e.g., P(AA or BB) = P(AA) + P(BB)).
All final probabilities sum to 1; for 'at least one', use 1 - P(none).
Conditional probabilities appear on later branches when outcomes depend on previous ones (e.g., without replacement).
In 'without replacement', denominators decrease by 1 on the second set of branches.

Two sets of traffic lights

Combined Probabilities

And means multiply: P(A and B) = P(A) × P(B).
Or means add: P(AA or BB) = P(AA) + P(BB).
Rephrase questions using 'and'/'or' to apply the rules.
Independent events do not affect each other (e.g., dice rolls); 'without replacement' events are not independent.
P(not A) = 1 - P(A) can simplify 'at least one' calculations.

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1.A two-way table shows the number of students in Year 12 and Year 13 studying Spanish or German. There are 25 Year 12 students (15 Spanish, 10 German) and 30 Year 13 students (5 Spanish, 25 German). What is the probability that a randomly selected student from the college studies Spanish?
Easy
- A\frac{20}{55}
- B\frac{15}{55}
- C\frac{5}{55}
- D\frac{35}{55}
2.In a Venn diagram, the universal set ℰ has 11 elements. Set A contains 5 elements, set B contains 4 elements, and the intersection A ∩ B contains 2 elements. What is P(A ∩ B)?
Easy
- A\frac{2}{11}
- B\frac{5}{11}
- C\frac{4}{11}
- D\frac{7}{11}
3.A tree diagram shows the probability of rain (1/3) and no rain (2/3). If it rains, Amira goes fishing with probability 3/5; if not, she goes with probability 3/4. What is the probability that on any day Amira goes fishing?
Medium
- A\frac{7}{10}
- B\frac{1}{3}
- C\frac{2}{5}
- D\frac{3}{5}
4.A bag contains 15 red beads and 10 yellow beads. Ariana picks a bead at random, records its colour, and replaces it. She then picks another bead at random. What is the probability that she picks two red beads?
Medium
- A\frac{9}{25}
- B\frac{3}{5}
- C\frac{2}{5}
- D\frac{3}{10}
5.ℰ = {odd numbers less than 30}. A = {3, 9, 15, 21, 27}. B = {5, 15, 25}. A number is chosen at random from ℰ. What is the probability that the number is in A ∪ B?
Medium
- A\frac{7}{15}
- B\frac{8}{15}
- C\frac{6}{15}
- D\frac{5}{15}
6.On any Saturday, the probability that Arun plays football is 3/4. The probability that Bob plays football is 2/5. Assuming independence, what is the probability that both play football on a Saturday?
Medium
- A\frac{3}{10}
- B\frac{6}{20}
- C\frac{1}{2}
- D\frac{7}{20}
7.Machine A makes bottles with a 0.02 probability of being faulty. Machine B makes bottles with a 0.05 probability of being faulty. One bottle is taken at random from each machine. What is the probability that at least one bottle is faulty?
Medium
- A0.069
- B0.001
- C0.07
- D0.0694
8.Two ordinary fair dice are rolled. What is the probability that both dice land on a number less than 3?
Easy
- A\frac{1}{9}
- B\frac{1}{6}
- C\frac{2}{3}
- D\frac{1}{3}

Mehr Fragen

9.The probability that the school bus is late is 9/10. If late, the probability Seb travels on the bus is 15/16. If on time, the probability Seb travels is 3/4. What is the probability that Seb travels on the bus?
Medium
- A\frac{141}{160}
- B\frac{135}{160}
- C\frac{147}{160}
- D\frac{129}{160}
10.Box A and box B each contain blue and green pens. P(Raphael picks blue from A) = 2/3. P(both pick blue) = 8/15. What is P(Paulo picks blue from B)?
Medium
- A\frac{4}{5}
- B\frac{2}{5}
- C\frac{8}{15}
- D\frac{16}{45}
11.What is the probability of rolling a 6 on a fair dice?
Easy
- A\frac{1}{6}
- B\frac{1}{2}
- C\frac{1}{3}
- D\frac{5}{6}
12.A bag contains 3 blue and 8 red counters. A counter is taken at random and replaced. What is the probability of taking a red counter?
Easy
- A\frac{8}{11}
- B\frac{3}{11}
- C\frac{8}{14}
- D\frac{5}{11}
13.Two events A and B are independent. P(A) = 0.3, P(B) = 0.4. What is P(A and B)?
Easy
- A0.12
- B0.7
- C0.1
- D0.58
14.In a class of 30 students, 15 study Spanish, 3 of the Spanish students also study German, and 7 students study neither language. What is the probability that a randomly selected student studies Spanish but not German?
Medium
- A\frac{12}{30}
- B\frac{15}{30}
- C\frac{3}{30}
- D\frac{8}{30}
15.In a Venn diagram, the universal set ℰ has 11 elements. Set A contains 5 elements, set B contains 4 elements, and the intersection A ∩ B contains 2 elements. What is the probability that a number chosen at random from ℰ is in A but not in B?
Medium
- A\frac{3}{11}
- B\frac{5}{11}
- C\frac{2}{11}
- D\frac{1}{11}
16.A tree diagram shows the probability of rain (1/3) and no rain (2/3). If it rains, Amira goes fishing with probability 3/5; if not, she goes with probability 3/4. What is the probability that Amira goes fishing on any day?
Medium
- A\frac{1}{3} \times \frac{3}{5} + \frac{2}{3} \times \frac{3}{4}
- B\frac{1}{3} \times \frac{3}{5}
- C\frac{2}{3} \times \frac{3}{4}
- D\frac{1}{3} \times \frac{3}{5} \times \frac{2}{3} \times \frac{3}{4}
17.On any Saturday, the probability that Arun plays football is 3/4. The probability that Bob plays football is 2/5. Assuming independence, what is the probability that both play football?
Medium
- A\frac{3}{4} \times \frac{2}{5}
- B\frac{3}{4} + \frac{2}{5}
- C\frac{3}{4} \times \frac{3}{4}
- D\frac{2}{5} \times \frac{2}{5}
18.Machine A makes bottles with a 0.02 probability of being faulty. Machine B makes bottles with a 0.05 probability of being faulty. One bottle is taken at random from each machine. What is the probability that at least one is faulty?
Medium
- A0.02 × 0.05
- B0.02 + 0.05 - 0.02×0.05
- C0.02 + 0.05
- D1 - 0.02×0.05
19.At a school, students can travel using one of three bus routes: A, B, and C. 40% of students use route A, 35% use route B, and the rest use route C. Buses on route A have an 8% chance of being late, route B have a 5% chance, and route C have a 12% chance. A student is chosen at random. What is the probability that they are late?
Hard
- A0.0795
- B0.0805
- C0.032
- D0.0875
20.In a class of 32 students, 5 do not study any languages, 15 study German, and 18 study Spanish. A student is chosen at random. What is the probability that the student studies Spanish but not German?
Medium
- A12/32
- B18/32
- C15/32
- D3/32
21.A bag contains 3 blue and 8 red counters. A counter is taken at random and not replaced. A second counter is then taken. What is the probability that both counters are red?
Medium
- A64/121
- B56/110
- C28/55
- D8/11
22.A soccer team plays two matches. The probability they win the first match is 0.6. If they win the first, the probability they win the second is 0.8; if they lose the first, the probability they win the second is 0.3. What is the probability they win exactly one match?
Medium
- A0.36
- B0.48
- C0.52
- D0.24
23.Box A contains 6 blue and 4 green pens. Box B contains 5 blue and 5 green pens. Raphael picks a pen at random from box A and Paulo picks a pen at random from box B. What is the probability that both pick a green pen?
Medium
- A0.20
- B0.25
- C0.16
- D0.40
24.A fair dice is rolled twice. What is the probability of getting a 6 on both rolls?
Easy
- A1/36
- B1/6
- C1/12
- D1/18
25.A bag contains 5 red and 3 blue marbles. A marble is taken at random, its colour noted, and then replaced. A second marble is taken. What is the probability that both marbles are red?
Medium
- A25/64
- B5/8
- C5/14
26.The two-way table shows the number of students in Year 12 and Year 13 studying Spanish or German. There are 25 Year 12 students (15 Spanish, 10 German) and 30 Year 13 students (5 Spanish, 25 German). A student is chosen at random from Year 13. What is the probability they study Spanish?
Easy
- A5/30
- B5/55
- C15/55
- D5/25
27.Two ordinary fair dice are rolled. What is the probability that the sum of the two numbers is 7?
Medium
- A1/6
- B5/36
- C1/9
- D1/12
28.Box A contains 3 blue and 2 green pens. Box B contains 4 blue and 1 green pen. Raphael picks a pen from box A and Paulo picks a pen from box B. What is the probability that both pick a blue pen?
Medium
- A12/25
- B3/5
- C4/5
- D7/25
29.A fair coin is flipped twice. What is the probability of getting two heads?
Easy
- A1/4
- B1/2
- C1/3
- D1/8
30.In a two-way table, what does each cell represent?
Easy
- AThe frequency of a combination of two categories
- BThe total number of observations
- CThe probability of an event
- DThe sum of row and column totals
31.The two-way table shows the number of students in Year 12 and Year 13 studying Spanish or German. How many students are in Year 13?
Easy
- A30
- B25
- C35
- D55
32.In a Venn diagram, the universal set ℰ has 11 elements. Set A contains 5 elements, set B contains 4 elements, and the intersection A ∩ B contains 2 elements. How many elements are in A ∪ B?
Medium
- A7
- B9
- C5
- D6
33.In a two-way table, the sum of all row totals equals the sum of all column totals.
Easy
True or false?
34.In a Venn diagram showing frequencies, what does the number in the intersection of two sets represent?
Easy
- AThe number of elements in both sets
- BThe number of elements in only one set
- CThe total number of elements in the universal set
- DThe number of elements in neither set
35.A bag contains 4 red and 6 blue marbles. One marble is drawn at random and replaced. Then a second marble is drawn. What is the probability that both marbles are blue?
Medium
- A0.36
- B0.24
- C0.6
- D0.16
36.In a Venn diagram, the universal set has 20 elements. Set A has 10, set B has 8, and A ∩ B has 3. Which of the following statements are correct? (Select all that apply)
Hard
- AP(A only) = 7/20
- BP(B only) = 8/20
- CP(A ∪ B) = 15/20
- DP(neither A nor B) = 10/20
37.Arrange the following steps in order to find the probability of at least one success in two independent trials.
Medium
- Identify the probability of success for each trial.
- Find the probability of failure for each trial (1 − P(success)).
- Multiply the failure probabilities together to find P(no success).
- Subtract P(no success) from 1 to find P(at least one success).
38.Match each description with the correct probability notation.
Easy
- Probability of A given B
- Probability of A and B
- Probability of A or B
- P(A \mid B)
- P(A \cap B)
- P(A \cup B)
39.When drawing a tree diagram for 'without replacement' scenarios, the probabilities on the second set of branches are the same as on the first set.
Easy
True or false?
40.A spinner has numbers 1 to 5, each equally likely. It is spun twice. What is the probability that the sum is at least 8?
Medium
- A6/25
- B4/25
- C7/25
- D3/25
41.Match each probability description with the correct calculation from a two-way table. The table shows 55 students: 25 in Year 12 (15 Spanish, 10 German) and 30 in Year 13 (5 Spanish, 25 German).
Medium
- P(Spanish and Year 12)
- P(Spanish)
- P(Spanish given Year 13)
- P(Year 12 given Spanish)
- 15/55
- 20/55
- 5/30
- 15/20
42.Arrange the following steps in the correct order to calculate the probability that a randomly selected student from a class of 30 students studies Spanish but not German, given that 15 study Spanish, 3 study both, and 7 study neither.
Hard
- Draw a Venn diagram with two overlapping circles labelled Spanish and German.
- Place 3 in the intersection (Spanish and German).
- Place 12 in Spanish only (15 - 3).
- Place 7 outside both circles.
- Calculate total inside circles: 30 - 7 = 23.
- Find German only: 23 - (12 + 3) = 8.
- Probability = number in Spanish only / total = 12/30.
43.Complete the sentence about probability tree diagrams.
Easy
The probabilities on each pair of branches from the same point add up to ____.
44.A two-way table shows the number of students in Year 10 and Year 11 who study French or German. There are 50 students in Year 10 and 60 in Year 11. 30 Year 10 students study French and 40 Year 11 students study German. How many students study French in total?
Easy
45.What is the probability of an event that is certain to happen? Give your answer as a decimal.
Easy
46.A bag contains 4 red and 6 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are the same colour? Give your answer as a fraction in simplest form.
Medium
47.Explain how to find the probability that a student selected at random from a class studies Spanish, given that the student is in Year 12, using a two-way table.
Medium
48.A two-way table shows the number of students in Year 10 and Year 11 who study French or German. There are 50 students in Year 10 and 60 in Year 11. 30 Year 10 students study French and 40 Year 11 students study German. How many students study French in total?
Easy
- A50
- B55
- C45
- D60
49.A bag contains 4 red and 6 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are the same colour? Give your answer as a fraction in simplest form.
Medium
- A6/15
- B8/15
- C9/15
- D7/15

Lernnotizen

## Probability Diagrams – Quick Revision ### 1. Two-Way Tables - Display frequencies for two categories. - Add row and column totals; overall total in bottom-right. - Probability = (number in region) / (total). - For conditional probability (e.g., given a row), denominator is the row total. ### 2. Venn Diagrams - Each region shows frequency or elements. - **Key notation:** - $A \cup B$: union (in A or B or both). - $A \cap B$: intersection (in both). - $A'$: complement (not in A). - Probability = (count in region) / (total). - Conditional: given in A, probability of B = $\frac{P(A \cap B)}{P(A)}$. ### 3. Probability Tree Diagrams - Branches show outcomes; probabilities on each branch. - **Rules:** - P(not A) = 1 – P(A). - Multiply along branches for "and". - Add separate cases for "or". - For "at least one", use 1 – P(none). - Conditional probabilities: second set of branches may differ (e.g., without replacement). ### 4. Combined Probabilities - **And rule:** P(A and B) = P(A) × P(B) (for independent events). - **Or rule:** P(A or B) = P(A) + P(B) (for mutually exclusive). - Rephrase questions using "and"/"or". - Without replacement: probabilities change after each draw. ### Key Formulas - $P(\text{not } A) = 1 - P(A)$ - $P(A \cap B) = P(A) \times P(B)$ (independent) - $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ - For tree: multiply branches, add final outcomes.

Two-way table: A table that displays frequencies for two categorical variables, with row and column totals.
Venn diagram: A diagram with overlapping circles inside a rectangle (universal set) showing relationships between sets.
Union (∪): The set of elements that are in A or B or both.
Intersection (∩): The set of elements that are in both A and B.
Complement ('): The set of elements not in the given set.
Tree diagram: A diagram with branches representing sequential events, with probabilities on each branch.
Conditional probability: The probability of an event given that another event has occurred; e.g., P(A|B).
Independent events: Events where the outcome of one does not affect the outcome of the other.
Combined probability: Probability of successive events, using multiplication for 'and' and addition for 'or'.

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