Question

2.15 A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

a)   A head on the first flip

b)   A tail on the second flip given that the first toss was a head

c)   Two tails

d)   A tail on the first and a head on the second

e)   A tail on the first and a head on the second or a head on the first and a tail on the second

f)   At least one head on the two flips

2.29 Which of the following are probability distributions? Why?

a) Random Variable X Probability

       2               0.1

       -1               0.2

       0               0.3

       1               0.25

       2               0.15

b) Random Variable Y Probability

       1               1.1

       1.5               0.2

       2               0.3

       2.5               0.25

       3               -1.25

c) Random Variable Z Probability

       1               0.1

       2               0.2

       3               0.3

       4               0.4

       5               0.0

2.32 David Upton is president of Upton Manufacturing, a producer of Go-Kart tires. Upton makes 1,000 tires per day with the following resources:

Labor:           400 hours per day @ $12.50 per hour

Raw material:       20,000 pounds per day @ $1 per pound

Energy:           $5,000 per day

Capital:           $10,000 per day

a)   What is the labor productivity per labor-hour for these tires at Upton Manufacturing?

b)   What is the multifactor productivity for these tires at Upton Manufacturing?

c)   What is the Percent change in multifactor productivity if Upton can reduce the energy bill by $1,000 per day without cutting production or changing other inputs.  

  

      

2.38 Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution.

a)   What is the probability that sales will be greater than 5,500 oranges?

b)   What is the probability that sales will be greater than 4,500 oranges?

c)   What is the probability that sales will be less than 4,900 oranges?

d)   What is the probability that sales will be less than 4,300 oranges?

2.43 Patients arrive at the emergency room of Costa Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

a)   Using Appendix C, compute the probability of exactly 0,1,2,3,4, and 5 arrivals per day.

b)   What is the sum of these probabilities, and why is the number less than 1?

2.45 Cars arrive at Cara’s Muffler shop for repair work at an average of 3 per hour, following an exponential distribution.

a)   What is the expected time between arrivals?

b)   What is the variance of the time between arrivals?

2.51 Use the F table in Appendix D to find the value of F for the upper 5% of the F distribution with:

a)   df1 = 5, df2 = 10

b)   df1 = 8, df2 = 7

c)   df1 = 3, df2 = 5

d)   df1 = 10, df2 = 4

Answer 1

2.15

Probability = Number of favourable outcomes/Total number of outcomes

Let H stand hor heads and T stand for tails

(i) In case of flipping a dollar the possible outcomes can be :[H,T]

P(H)=1/2

(ii) Possible outcomes: [H H ,H T]
P(T)=1/2

(iii) Possible outcomes: [ HH, TH, HT ,TT]
P(T,T)=1/4

(iv) Possible outcomes: [HH, TH, HT,TT]

P(T,H)=1/4

(v) Possible outcomes: [HH, TH, HT,TT]  

P(T,H)=1/4

P(H,T)=1/4

P(T,H or H,T)=1/4+1/4=1/2

(vi) P(T,H) +P(H,T) +P(H,H)+P(T,T)=1
Probability of atleast one head = 1-P(T,T) = 1-1/4 = 3/4