Question

2. QUESTION:

A fair coin is tossed, and a fair die is thrown. Write down sample spaces for

(a) the toss of the coin;

(b) the throw of the die;

(c) the combination of these experiments.

Let A be the event that a head is tossed, and B be the event that an odd number is thrown. Directly from the sample space, calculate P(A ∩ B) and P(A ∪ B).

Answer 1

(a) $\{\text{Head},\text{Tail}\}$

(b) $\{1,2,3,4,5,6\}$

(c) $\{(1\cap\text{Head}), (1\cap\text{Tail}),\dots,(6\cap\text{Head}), (6\cap\text{Tail})\}$

Clearly $\mathbb{P}(A)=\frac{1}{2}=\mathbb{P}(B)$. We can assume that the two events are independent, so $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)=\frac{1}{4}$. Alternatively, we can examine the sample space above and deduce that three of the twelve equally likely events comprise $A\cap B$. Also, $\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)=\frac{3}{4}$, where this probability can also be determined by noticing from the sample space that nine of twelve equally likely events comprise $A\cup B$.