Question

A company surveyed 1000 people on their age and the number of jeans purchased annually. The results of the poll are shown in the table.

	Jeans Purchased Annually
	0	1	2	3 or More	Total
Under 12	0	50	61	79	190
12-18	26	54	66	44	190
19-25	36	66	117	41	260
over 25	73	58	211	18	360
Total	135	228	455	182	1000

A person is selected at random. Compute the probability of the following.

(a) The person is over 25 and purchases 3 or more pairs of jeans annually.
  

(b) The person is in the age group 12-18 and purchases at most 2 pairs of jeans annually.
  

(c) The person is in the age group 19-25 or purchases no jeans annually.
  

(d) The person is older than 18 or purchases exactly 1 pair of jeans annually.
  

Answer 1

    JEANS PURCHASED ANNUALLY

Under	0(X)	1(Y)	2(Z)	3 or more(T)	Total
12(A)	0	50	61	79	190
12-18(B)	26	54	66	44	190
19-25(C)	36	66	117	41	260
over 25(D)	73	58	211	18	360
Total	135	228	455	182	1000

A person is selected at random. Before solving the problem, let us make some assumptions such as "A", "B", "C", "D" and "X", "Y", "Z", "T" to denote each category of the persons and the purchases categories respectively. Observe the data shown in the above table.

(a) The Probability of a selected person is over 25 and purchases 3 or more pairs of jeans annually is represented mathematically as P(D T) = P(D) and P(T).

That is, The Probability of a selected person is over 25 and purchases 3 or more pairs of jeans annually is the product of the probality of a selected person is over 25 and probality of the purchases 3 or more.

P(D T) = P(D)*P(T) = (0.36)x(0.182) = 0.06552

b) The person is in the age group 12-18 and purchases at most 2 pairs of jeans annually.

"Probability of purchasing at most 2 pairs of jeans annually" includes probability of purchasing 0 or 1 or 2 but not more than 2. It is represented mathematically as P(X Y Z ) . Instead of finding P(X Y Z ) directly, we can find its value in another way.

P(X Y Z ) = 1 - P(T) = 1- 0.36 = 0.64 { we have found the value of P(T) in the previous subdivision}

Probalility of the person is in the age group 12 -18 is denoted as P(B).

From the table, we find that .

Probability of the person is in the age group 12 -18 and purchases at most 2 pairs of jeans annually is represented by P((B) (X Y Z )) = P(B) * P(X Y Z ) = (0.19) x (0.64) = 0.1216.

c) The person is in the age group 19-25 or purchases no jeans annually.

The probability of the person is in the age group 19-25 is denoted as P(C) and the probability of purchases of no jeans annually is denoted as P(X).

From the above table, we could find that probability of the person is in the age group 19-25 is

and the probability of purchasing of no jeans annually is

The probability of the person is in the age group 19-25 or purchases no jeans annually is denoted as P(C X).

P(CUX) = P(C) + P(X) - P(C   X) ---------(1)

P(CX) = P(C) * P(X) = (0.26) x (0.135) = 0.0351

Entering the values in the equation (1), we get

P(CUX) = (0.26) + (0.135) - (0.0351) = (0.0395) - (0.0351) =0.0044

(d) The person is older than 18 or purchases exactly 1 pair of jeans annually.

Probability of the person is older than 18 is denoted as P(CD).

The value of P(CD) is P(CD) = P(C) * P(D) = (0.26) x (0.36) = 0.0936 (Go through the above sub divisions for the values of P(C) and P(D).

Probability of purchasing exactly 1 pair of jeans annually is denoted as P(Y )

Probability of the person is older than 18 or purchases exactly 1 pair of jeans annually is denoted mathematically as

P((CD)UY) = P(CD) + P(Y) - P(CD Y) = (0.0936) + (0.228) -((0.26) x (0.36) x (0.228)) = (0.3216) - (0.0213)

= 0.3003