Mathematics (Core)Exam code: 0580
Set Notation and Probability Diagrams
Probability
The two rules
- Where two things happen one after the other, two rules cover almost everything
- "And" means multiply, so the probability of A happening and then B is P(A) × P(B), where P(A) is short for the probability of A
- "Or" means add, so the probability of one case or another, when the two cannot both happen, is the sum of their probabilities
- Rephrasing a question in terms of "and" and "or" first is what turns it into a calculation
- "Both counters are red" is red and red
- "One of each colour" is red and blue or blue and red
- The probability of all the possible cases together is 1, which is a quick check on any full list
- Where a question asks for at least one, it is usually faster to work out the probability of none and subtract from 1
Independent events
- Two events are independent when the first does not change the second
- With replacement means the item goes back, so the bag is the same for the second pick and the two probabilities are equal
- These papers combine events with replacement only, so the result of the first stage never changes the probabilities for the second
- Keeping fractions unsimplified while working makes them far easier to add at the end
Worked example
Two picks with replacement
A bag holds 5 red counters and 7 blue counters. A counter is taken at random, its colour is recorded, and it is put back. A second counter is then taken. Find the probability that both are red.
Solution:
- There are 5 + 7 = 12 counters, so the first pick is red with probability 5/12
- The counter goes back, so the bag is unchanged and the second pick is red with probability 5/12 as well
- "And" means multiply, so P(red then red) = 5/12 × 5/12
- P(red then red) = 25/144