Probability Diagrams

Probability

Flashcards

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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)
- "Or" means add, so the probability of one case or another 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