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Dependent Events Solutions

• When the occurrence of the first event changes the probability of the subsequent events, the events are    dependent   .

• The    Multiplication Rule for Dependent Probability    states: When A and B are dependent events, then $\mathrm{P}\left(\mathrm{A} \mathrm{and} \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}\right)·\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)$.

• The phrase    “without replacement”    often represents dependent probability.

• Not replacing an element changes the    total number of outcomes   .

Multiplication Rule for:
Independent Probability $\mathrm{P}\left(\mathrm{A} \mathrm{and} \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}\right)·\mathrm{P}\left(\mathrm{B}\right)$	Dependent Probability $\mathrm{P}\left(\mathrm{A} \mathrm{and} \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}\right)·\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)$

Example 1

A jar contains 11 red, 27 yellow, 16 blue, and 6 green marbles. Beau selects three marbles, one after the other, without replacement. If the marbles were selected randomly, what is the probability that Beau selects a yellow, then another yellow, and then a blue marble? Round to the nearest tenth of a percent.

$\mathrm{P}\left(\mathrm{Y}, \mathrm{Y}, \mathrm{B}\right)=\mathrm{P}\left(\mathrm{Y}\right)·\mathrm{P}\left(\mathrm{Y}|\mathrm{Y}\right)·\mathrm{P}\left(\mathrm{B}|\mathrm{Y}, \mathrm{Y}\right)$

$\begin{array}{rcl}\mathrm{Total}: 11+27+16+6& =& 60\\ \mathrm{P}\left(\mathrm{Y}, \mathrm{Y}, \mathrm{B}\right)& =& \frac{27}{60}·\frac{26}{59}·\frac{16}{58}=0.0547\\ \mathrm{P}\left(\mathrm{Y}, \mathrm{Y}, \mathrm{B}\right)& =& 5.5\%\end{array}$

Example 2

Meredith checked out four graphic novels, three mysteries, and three biographies from the library. What is the probability that the first four books Meredith randomly selects are a mystery, a graphic novel, a graphic novel, and a biography?

$\begin{array}{rcl}\mathrm{P}\left(\mathrm{m}, \mathrm{g}, \mathrm{g}, \mathrm{b}\right)& =& \frac{3}{10}·\frac{4}{9}·\frac{3}{8}·\frac{3}{7}\\ & =& \frac{3}{10}·\frac{\cancel{4}}{\cancel{9}}·\frac{\cancel{3}}{{\cancel{8}}_2}·\frac{\cancel{3}}{7}=\frac{3}{140}=0.0214\end{array}$

Meredith has about a 2% chance of selecting (m, g, g, b) for her books.