MathematicsExam code: 0580
Probability Diagrams
Probability
Combining Two Events
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), using the probability of B once A has happened
- "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
With and without replacement
- With replacement means the item goes back, so the second probability is the same as the first and the events are independent
- Without replacement means it does not, so the totals change and the second probability depends on what happened first
- Both the top and the bottom of the second fraction can change: the bottom always drops by one, and the top drops by one only for whatever was taken
- Keeping fractions unsimplified while working makes them far easier to add at the end
Worked example
With and without replacement
A bag holds 5 red counters and 7 blue counters. Two counters are taken. Find the probability that both are red (a) with replacement, (b) without replacement.
Solution:
- There are 5 + 7 = 12 counters, so the first pick is red with probability 5/12
- (a) With replacement, the counter goes back, so nothing changes for the second pick
- P(red then red) = 5/12 × 5/12 = 25/144
- (b) Without replacement, one red has gone, so 4 reds remain out of 11 counters
- The denominator drops by one because a counter was removed; the numerator drops only because the one removed was red
- P(red then red) = 5/12 × 4/11 = 20/132 = 5/33
- The second answer is smaller, which makes sense: removing a red makes a second red less likely
Exam tip
Two picked from a group means without replacement
Picking without replacement is set in 9 of the 21 papers, and the question does not always use those words: two people or objects chosen from a group cannot be the same one twice, so the second fraction is out of one fewer. Working the other kind of pick (with replacement for without, or the reverse) usually scores one mark at most out of three or four, so decide which kind it is before the first fraction.