Section 1.2 Suppose A 1, A 2,..., A k, are k events. The k events are called mutually exclusive if The k events are called mutually exhaustive if A i  A j =  whenever i  j. A 1  A 2 ...  A k = S the outcome space. Important definitions and theorems in the text: The definition of probability. Definition P(A) = 1 – P(A / ) Theorem P(  ) = 0 Theorem If A  B, then P(A)  P(B) Theorem For each event A, P(A)  1 Theorem If A and B are any two events, then P(A  B) = P(A) + P(B) – P(A  B) Theorem 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. (a) (b) (c) (d) Find the outcome space for each of the random variables defined, and indicate whether or not the outcomes are equally likely. 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. 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. 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. 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. (a) (b) The number facing upward is observed when a fair 20-sided die is rolled once. Define following events: A 1 = the number is a perfect square = {1, 4, 9, 16} A 2 = there is no number = { } =  A 3 = the number is 9 = {9} A 4 = the number is odd = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} A 5 = the number is prime = {2, 3, 5, 7, 11, 13, 17, 19} 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. Find P(A 1 ), which is the probability that the number is a perfect square, and find P(A 1 / ), which is the probability that the number is not a perfect square. P(A 1 ) =4/20 = 1/5P(A 1 / ) = 1  1/5 = 4/5

(c)Find P(A 2 ), which is the probability that there is no number. P(A 2 ) = P(  ) = 0 (d)Find P(A 3 ), which is the probability that the number is 9. P(A 3 ) =1/20 (e)Find P(A 4 ), which is the probability that the number is odd. P(A 4 ) =10/20 = 1/2 (f)Find P(A 5 ), which is the probability that the number is prime. P(A 5 ) =8/20 = 2/5 (g) Find P(A 1  A 5 ), which is the probability that the number is either a perfect square or a prime. P(A 1  A 5 ) = P(A 1 ) + P(A 5 ) = (h) Find P(A 4  A 5 ), which is the probability that the number is either odd or a prime. 4/20 + 8/20 = 12/20 = 3/5 P(A 4  A 5 ) =P(A 4 ) + P(A 5 )  P(A 4  A 5 ) = 10/20 + 8/20  7/20 = 11/20