With replacement, the probabilities stay the same on the second branches. Without replacement, they change, because there is one fewer item. A bag with 5 red and 3 blue counters: P(red then red) without replacement = 5/8 × 4/7 = 20/56 = 5/14.
For P(both the same colour), add the red–red and blue–blue paths: 5/8 × 4/7 + 3/8 × 2/7 = 20/56 + 6/56 = 26/56 = 13/28. Multiply along branches and add between paths.
Worked example
A bag has 6 red and 2 blue counters. Two are taken without replacement. Find P(both blue).
- First blue: 2/8. Second blue: 1/7.
- 2/8 × 1/7 = 2/56.
Answer:1/28
Key words
- with replacement
- the item is put back
- without replacement
- the item is kept out
- dependent events
- one affects the other
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Common Core-referenced: HSS-CP.B.8