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).


1
Expert's answer
2022-06-21T12:27:53-0400

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

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

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


Clearly P(A)=12=P(B)\mathbb{P}(A)=\frac{1}{2}=\mathbb{P}(B). We can assume that the two events are independent, so P(AB)=P(A)P(B)=14\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 ABA\cap B. Also, P(AB)=P(A)+P(B)P(AB)=34\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 ABA\cup B.


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