Question

(Probability Distribution)

1. If one ball each is drawn from 3 boxes, the first containing 3 red, 2 yellow, and 1 blue, the second box contains 2 red, 2 yellow, and 2 blue, and the third box with 1 red, 4 yellow, and 3 blue. What is the probability that all 3 balls drawn are different colors?
2. In a viral pool test it is known that in a group of five (5) people, exactly one (1) will test positive. If they are tested one by one in random order for confirmation, what is the probability that only two (2) tests are needed?

Answer 1

1.

$$P(3\ different)=P(RYB)+P(RBY)$$

$$+P(YRB)+P(YBR)+P(BYR)+P(BRY)$$

$$=\dfrac{3}{6}(\dfrac{2}{6})(\dfrac{3}{8})+\dfrac{3}{6}(\dfrac{2}{6})(\dfrac{4}{8})+\dfrac{2}{6}(\dfrac{2}{6})(\dfrac{3}{8})$$

$$+\dfrac{2}{6}(\dfrac{2}{6})(\dfrac{1}{8})+\dfrac{1}{6}(\dfrac{2}{6})(\dfrac{1}{8})+\dfrac{1}{6}(\dfrac{2}{6})(\dfrac{4}{8})$$

$$=\dfrac{17}{72}$$

2.

The first test must be negative

$$P(Negative)=\dfrac{5-1}{5}$$

The second test must be positive

$$P(Positive)=\dfrac{1}{5-1}$$

Then

$$P(X=2)=P(1^{st}Negative, 2^{nd}Posirive)$$

$$=\dfrac{5-1}{5}(\dfrac{1}{5-1})=\dfrac{1}{5}$$