Question

Three fair dices are rolled together. First dice Shows odd number, second dice shows 1 or 3 and whether third dice shows a 5 or 6- check the events are. independent or not.

Answer 1

There are $6^3=216$ outcomes for rolling three fair dices together.

Suitable outcomes

$$(1, 1, 5), (1, 1, 6), (1, 3, 5), (1, 3, 6),$$

$$(3, 1, 5), (3, 1, 6), (3, 3, 5), (3, 3, 6),$$

$$(5, 1, 5), (5, 1, 6), (5, 3, 5), (5, 3, 6)$$

The probability of the event "odd number on first and 1 or 3 on second, and 5 or 6 on third" is

$$P(I\cap II\cap III)=\dfrac{12}{216}=\dfrac{1}{18}$$

The probability of the event "odd number on first" is

$$P(I)=\dfrac{3}{6}=\dfrac{1}{2}$$

The probability of the event "1 or 3 on second" is

$$P(II)=\dfrac{2}{6}=\dfrac{1}{3}$$

The probability of the event "5 or 6 on third" is

$$P(III)=\dfrac{2}{6}=\dfrac{1}{3}$$

Hence

$$P(I)P(II)P(III)=\dfrac{1}{2}(\dfrac{1}{3})(\dfrac{1}{3})=P(I\cap II\cap III)$$

 The events are independent.