Question

Bob suddenly remembers that today is his girlfriend’s birthday, and rushes into a nearby florist, to buyher some flowers. There he finds a large urn containing a population of three equally-divided types of differently-colored roses, at different prices: yellow roses cost $1.10 each, pink roses cost $3.30 each, and red roses cost $4.60 each. As he is in a hurry, he simply decides to select a dozen roses at random.

(a) What are the minimum and maximum costs of the dozen roses?

(b) Calculate the expected cost of a single rose, and use it calculate the expected cost of the dozen roses.

(c) Calculate the variance in the cost of a single rose.

(d) Calculate the approximate probability that Bob will have to pay between $30 and $45 for the dozen roses.

Answer 1

a)

Minimum cost = 1.1 * 12 = 13.2

Maximum cost = 4.6 * 12 = 55.2

b)

X	P(X)	X*P(X)	X² * P(X)
1.1	0.3333	0.367	0.403
3.3	0.3333	1.100	3.630
4.6	0.3333	1.533	7.053

	P(X)	X*P(X)	X² * P(X)
total sum =	1	3	11.09

mean = E[X] = Σx*P(X) =            3

Expected cost = 3*12 = 36

c)

E [ X² ] = ΣX² * P(X) =            11.0867
          
variance = E[ X² ] - (E[ X ])² =            2.0867

d)

µ =    36                              
σ =    5.003998401                              
we need to calculate probability for ,                                  
P (   30   < X <   45   )                  
=P( (30-36)/5.00399840127871 < (X-µ)/σ < (45-36)/5.00399840127871 )                                  
                                  
P (    -1.199   < Z <    1.799   )                   
= P ( Z <    1.799   ) - P ( Z <   -1.199   ) =    0.9640   -    0.1153   =    0.8487

Please revert back in case of any doubt.

Please upvote. Thanks in advance.