A taste test for a new coffee brand was performed, with results recorded in the

Question

A taste test for a new coffee brand was performed, with results recorded in the probability model below. All 4 table probabilities sum to 100% but two of these probabilities (shown here as x and y) were accidently lost.

                    Enjoyed Taste?

                    Yes                 No

Male            x                     18.4%
Female        32%                     y

Let A = "Did enjoy the taste"
Let B= "Is female"

Using this given table (which should also be shown in your write-up):

a) Find P (A). Also, find P (B). [These answers will include numbers along with symbols x and y.]

b) Find P (A and B). [Hint: Find this single probability value by direct-inspection of the table.]

c) Write an equation showing these four table values added together, and equaling their required sum.

d) If event A is independent of event B, then the product of your answers in part (a) must equal your answer in part (b). Can we find percentage values for x and y that solve this equation. If so, find all such solutions. If not, say why not. [Jint: Try using part (c), to make a variable substitution.]

Explanation / Answer

a) P(A) = P(enjoy the taste ) = P(yes) = 0.32 +x/100

P(B) = P(female ) = 0.32 + y/100

b) P(A and B) = P(female enjoying the taste ) = 0.32

c) x + 18.4 + 32 + y =100

x+y = 49.4

d) for independent event

P(AB) = P(A)* P(B)

0.32 = ( 0.32 +x/100)( 0.32 + y/100)

= 0.1024 + 0.32(x+y)/100 + xy/100*100

2176 = 32(x+y) +xy

2176 = 32 (49.6) +xy

2176 = 1587.2 + xy

588.8 = xy

as we know

(x+y)2 - (x- y)2 = 4xy

(x+y)2 -4xy = (x- y)2

(49.4)2 - 4 (588.8) = 85.16

x -y = 9.23

x+y = 49.4

x = 29.315%

y = 20.085%

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