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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 $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cup B)$.
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(a) $\{$ Head, Tail $\}$

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

(c) $\{(1 \cap H e a d),(1 \cap$ Tail $), \ldots,(6 \cap$ Head $),(6 \cap$ Tail $)\}$

Clearly $\mathrm{P}(A)=\frac{1}{2}=\mathrm{P}(B)$. We can assume that the two events are independent, so
$$\mathrm{P}(A \cap B)=\mathrm{P}(A) \mathrm{P}(B)=\frac{1}{4}$$
Alternatively, we can examine the sample space above and deduce that three of the twelve equally likely events comprise $A \cap B$. Also, $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A \cap B)=\frac{3}{4}$, where this probability can also be determined by
noticing from the sample space that nine of twelve equally likely events comprise $A \cup B$.
by Platinum (130,522 points)

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