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Section 1.2
Suppose A1, A2, ..., Ak, are k events.
The k events are called mutually exclusive if
Ai  Aj =  whenever i  j.
The k events are called mutually exhaustive if
A1  A2  ...  Ak = S the outcome space.
Important definitions and theorems in the text:
Definition 1.2-1
The definition of probability.
Theorem 1.2-1
P(A) = 1 – P(A/)
Theorem 1.2-2
P() = 0
Theorem 1.2-3
If A  B, then P(A)  P(B)
Theorem 1.2-4
For each event A, P(A)  1
If A and B are any two events, then P(AB) = P(A) + P(B) – P(AB)
Theorem 1.2-5
If A, B, and C are any three events, then P(ABC) =
P(A) + P(B) + P(C) – P(AB) – P(AC) – P(BC) + P(ABC)
Theorem 1.2-6
1. Find the outcome space for each of the random variables defined,
and indicate whether or not the outcomes are equally likely.
(a) The random variable X is defined to be the number of spots facing
upward when a fair die is rolled once.
{1, 2, 3, 4, 5, 6}
The outcomes are equally likely.
(b) The random variable X is defined to be the total number of spots
facing upward when two fair dice are each rolled once.
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} The outcomes are not equally likely.
(c) The random variable X is defined to be the number of heads facing
upward when a fair penny is tossed once.
{0, 1}
The outcomes are equally likely.
(d) The random variable X is defined to be the number of heads facing
upward when 10 fair pennies are each tossed once.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
The outcomes are not equally likely.
2. The number facing upward is observed when a fair 20-sided die is
rolled once. Define following events:
A1 = the number is a perfect square = {1, 4, 9, 16}
A2 = there is no number = { } = 
A3 = the number is 9 = {9}
A4 = the number is odd = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
A5 = the number is prime = {2, 3, 5, 7, 11, 13, 17, 19}
(a) Find the outcome space for this experiment, and indicate whether
or not the outcomes are equally likely.
{1, 2, 3, …, 20}
The outcomes are equally likely.
(b) Find P(A1), which is the probability that the number is a perfect
square, and find P(A1/ ), which is the probability that the number is
not a perfect square.
P(A1) = 4/20 = 1/5
P(A1/ ) = 1  1/5 = 4/5
(c) Find P(A2), which is the probability that there is no number.
P(A2) = P() = 0
(d) Find P(A3), which is the probability that the number is 9.
P(A3) = 1/20
(e) Find P(A4), which is the probability that the number is odd.
P(A4) = 10/20 = 1/2
(f) Find P(A5), which is the probability that the number is prime.
P(A5) = 8/20 = 2/5
(g) Find P(A1  A5), which is the probability that the number is either a
perfect square or a prime.
P(A1  A5) = P(A1) + P(A5) = 4/20 + 8/20 = 12/20 = 3/5
(h) Find P(A4  A5), which is the probability that the number is either
odd or a prime. P(A4  A5) = P(A4) + P(A5)  P(A4  A5) =
10/20 + 8/20  7/20 = 11/20