Statistics / Unit 6: Probability
6.6 Tree Diagrams and Sampling Without Replacement
Notes
A tree diagram lays out the stages of a probability problem. Multiply along branches for "and" and add across branches for "or".
- When drawing without replacement, probabilities change after each draw, so the draws are not independent.
- With replacement, the draws are independent and probabilities stay the same.
- Example: A bag has 5 red and 3 blue marbles. $\displaystyle P(\text{both red}) = \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.
Practice
A bag has 5 red and 3 blue marbles. Two are drawn without replacement. What is the probability that both are red?
$\displaystyle \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.
For a bag (5 red, 3 blue, drawing 2 without replacement), what is the probability the two marbles are different colors?
$\displaystyle \frac{5}{8} \cdot \frac{3}{7} + \frac{3}{8} \cdot \frac{5}{7} = \frac{15}{56} + \frac{15}{56} = \frac{30}{56} = \frac{15}{28}$.
A bag has 5 red and 3 blue marbles. You draw 2, replacing the first before the second draw. The probability both are red is:
With replacement the draws are independent: $\displaystyle \frac{5}{8} \cdot \frac{5}{8} = \frac{25}{64}$.
Ready for more? Take the Unit 6 test.