Question

Determine the probability of rolling a die, and it landing on a 6, and a ball on a roulette wheel landing on an even number (the roulette wheel contains the numbers 1 to 36, ignore zero and double zero) State whether the events are dependent or independent.

Answer 1

$$A = \{\text{ball on a roulette wheel landing on } 6\}$$ $$B = \{\text{ball on a roulette wheel landing on an even number}\} = \{\text{ball on 2, (or) 4, 6, 8, ..., 30, 32, 34, 36}\}$$ $$P(A) = \frac{1}{36}$$ $$P(B) = \frac{18 \text{ (because ball can be on 18 numbers)}}{36} = \frac{1}{2}$$

A finite set of events is mutually independent if every event is independent of any intersection of the other events.[3] That is, if for every subset $\{A_n\}$

$$P\left(\bigcap_{i=1}^{n} A_i\right) = \prod_{i=1}^{n} P(A_i)$$

This is called the multiplication rule for independent events.

Two events $A$ and $B$ are independent if their joint probability equals the product of their probabilities:

$$P(A \cap B) = P(A)P(B)$$

We have: $A \cap B = \{\text{ball on a roulette wheel landing on } 6\} = A$

$$P(A \cap B) = P(A) = \frac{1}{36} \neq \frac{1}{72} = P(A) * P(B); \quad P(A \cap B) \neq P(A) * P(B), \text{ so } A \text{ and } B \text{ are dependent.}$$