Question #351046

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) "\\{HEAD,TAIL\\}"

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

(c)

"\\{(HEAD,1), (HEAD,2), (HEAD,3),\\\\ (HEAD,4), (HEAD,5), (HEAD,6),\\\\\n (TAIL,1), (TAIL,2), (TAIL,3),\\\\ (TAIL,4), (TAIL,5), (TAIL,6)\\}"

The sample space of combination of the experiments has "12" elements with equal probabilities "\\frac{1}{12}."

"A\\cap B=\\{(HEAD,1), (HEAD,3), (HEAD,5)\\}",

"|A\\cap B|=3", hence "P(A\\cap B)=3\\cdot \\frac{1}{12}=\\frac{1}{4}."

"A\\cup B=\\{(HEAD,1), (HEAD,2), (HEAD,3),\\\\ (HEAD,4), (HEAD,5), (HEAD,6),\\\\\n(TAIL,1), (TAIL,3), (TAIL,5)\\},"

"|A\\cup B|=9," hence "P(A\\cup B)=9\\cdot \\frac{1}{12}=\\frac{3}{4}."

Answer: "P(A\\cap B)=\\frac{1}{4}, P(A\\cup B)=3\\cdot \\frac{1}{12}=\\frac{3}{4}."

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