Probability Diagrams
边玩边学
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课程笔记
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.
幻灯片
练习题
免费预览——49 题中的 8 题。注册即可查看全部。
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}