Question #351002

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).

Expert's answer

(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".

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