Let events A and B be the following. A: The Dow jones average rises on the first

Question

Let events A and B be the following. A: The Dow jones average rises on the first day. B: The Dow Jones average rises on the second day. a. Show that (A B) (A B) = B, b. Show that A (A B) = A B. Florin Frenti operates a small, used car lot that has three Mercedes (M_1, M_2, M_3) and two Toyotas (T_1, T_2). Two customers, Cezara and Anda, come to his lot, and each selects a car. The customers do not know each other, and there is no communication between them. Let the events A and B be defined as follows: A: The customers select at least one Toyota. B: The customers select two cars of two cars of the same model. a. Identify all pairs of cars in the sample space. b. Define event A. c. Define event B. d. Define the complement of A. e. Show that (A B) (A B) = B. f. Show that A (A B) - A B.

Explanation / Answer

(A)

All pair of cars in sample space

(M1, M2) (M1, M3) (M1, T1) (M1, T2)

(M2, M1) (M2, M3) (M2, T1) (M2, T2)

(M3, M1) (M3, M2) (M3, T1) (M3, T2)

(T1, M1) (T1, M2) (T1, M3) (T1, T2)

(T2, M1) (T2, M2) (T2, M3) (T2, T1)

total 20 pairs

(B)

Event A: The customer select at least one Toyota

(M1, T1) (M1, T2)

(M2, T1) (M2, T2)

(M3, T1) (M3, T2)

(T1, M1) (T1, M2) (T1, M3) (T1, T2)

(T2, M1) (T2, M2) (T2, M3) (T2, T1)

n(A) = 14

(C)

Event B: the customers select two cars of the same model

(M1, M2) (M1, M3)

(M2, M1) (M2, M3)

(M3, M1) (M3, M2)

(T1, T2)

(T2, T1)

n(B) = 8

(D)

complement of A: the customers select no Toyota

(M1, M2) (M1, M3)

(M2, M1) (M2, M3)

(M3, M1) (M3, M2)

n(not A) = 6

(E)

(A and B) = { (T1,T2) (T2, T1) }

(not A and B) = { (M1, M2) (M1, M3) (M2, M1) (M2, M3) (M3, M1) (M3, M2) }

(A and B) or (not A and B) = { (T1,T2) (T2, T1) (M1, M2) (M1, M3) (M2, M1) (M2, M3) (M3, M1) (M3, M2)  }

B = { (T1,T2) (T2, T1) (M1, M2) (M1, M3) (M2, M1) (M2, M3) (M3, M1) (M3, M2)  }

Hence proved

(F)

A or (not A and B) = { (M1, M2) (M1, M3) (M2, M1) (M2, M3) (M3, M1) (M3, M2) (M1, T1) (M1, T2) (M2, T1) (M2, T2) (M3, T1) (M3, T2) (T1, M1) (T1, M2) (T1, M3) (T1, T2) (T2, M1) (T2, M2) (T2, M3) (T2, T1) }

(A or B) =  { (M1, M2) (M1, M3) (M2, M1) (M2, M3) (M3, M1) (M3, M2) (M1, T1) (M1, T2) (M2, T1) (M2, T2) (M3, T1) (M3, T2) (T1, M1) (T1, M2) (T1, M3) (T1, T2) (T2, M1) (T2, M2) (T2, M3) (T2, T1) }

Hence proved

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