Answer to Question #186970 in Statistics and Probability for Anthonette

Question

D. Determine the possible values and construct the probability distribution for the random

variables described in each of the following situations. Draw the corresponding histogram for each

probability.

1. Two balls are drawn in succession without replacement from an urn containing 5 red balls and

6 blue balls. Let Z be the random variable representing the number of blue balls. Construct the probability distribution of the random variable Z.

Possible Outcome

Value of the Random

Variable Z

Value of the Random

Variable z

Probability P(Z)

Answer 1

Possible outcomes:

Z=0 - Two red balls were pulled out of the first urn;

Z=1 - A blue and a red ball was pulled out of the first urn;

Z=3 - Two blue balls were pulled out of the first urn;

Total 11 balls in the urn

The probability that there will be 0 blue balls (both balls are red): $p(Z=0) = \frac{5}{{11}} \cdot \frac{4}{{10}} = \frac{2}{{11}}$

The probability that there will be 1 blue ball (and one red ball): $p(Z=1) = \frac{5}{{11}} \cdot \frac{6}{{10}} + \frac{6}{{11}} \cdot \frac{5}{{10}} = \frac{6}{{11}}$

The probability that there will be 2 blue balls: $p(Z=2) = \frac{6}{{11}} \cdot \frac{5}{{10}} = \frac{3}{{11}}$

We have a series of probability distributions:

$\begin{matrix} Z&0&1&2\\ p&{\frac{2}{{11}}}&{\frac{6}{{11}}}&{\frac{3}{{11}}} \end{matrix}$

The corresponding histogram is: