# A set of questions on distribution of data

1. A coin is flipped twice.

a. What is the probability of two heads?

b. What is the probability of a head in the second toss?

c. What is the probability of a head in the second toss given that a head was obtained in the first toss?

d. At least one head in both flips.

2. The police placed three speed radars at three different locations (Palmyra, Dawson, and Slappey). The radars only work 40%, 30%, and 20% of the time. When you come to class late, the probabilities driving through the radar locations are 0.2, 0.3, and 0.5 respectively.

a. What is the probability of you getting a ticket?

b. Given that you got a ticket, what is the probability that you got it at Dawson?

3. Imagine that you will have an interview for your dream job in Los Angeles. Since you do not want to miss the interview,you bought two tickets to LA. One withDelta and the other with US Airways. The probability of a Delta cancellation is 5% and the probability of a US Airways cancellation is 7%. Assuming that you arrived to the airport on time, what is the probability that you arrive to the interview?

4. An oil driller company ventures in to various locations and their success or failure is independent from one location to another. Suppose the probability of a success is 0.20. The driller is planning to drill 10 locations.

a. What is the probability that a driller finds 2 successes?

b. The driller feels that he will go bankrupt if he does not have any successes. What is the probability that the company goes to bankrupt?

c. The company's share will duplicate its value if the number of successes is between 4 and 8 (both included). What is the probability that the share value will duplicate?

5. A recent study by the American Highway Patrolman's association reveals that 60% of American drivers use their seat belts. A sample of 15 drivers on interstate 75 is selected.

a. What is the probability that 6 drivers are using seatbelts?

b. What is the probability that between 5 and 7 (both included) are wearing seat belts?

c. What is the probability that more than 8 drivers were wearing seat belts?

6. 584 commercial flights arrive every day to Pittsburgh international airport. In a regular day the first fly arrives at 7:00 am and the last one at midnight. Assuming that the number of arrivals per hour is governed by a Poisson distribution, what is the probability that no plane arrives in the next 2 minutes.

7. A local drugstore owner knows that on average, 100 people per hour stop at the store.

a. Find the probability that in a given 3 minute period nobody enters in the store.

b. Find the probability that in a given 3 minute period more than 5 people enter the store?

8. A national geographic photographer invites you to work in a documentary about hammer sharks in the Pacific Ocean. Based on his last experience he expects to observe 10 sharks every hour.

a. What is the probability that you can observe 3 sharks in the next 30 minutes?

b. You get bored at the boat and you bet with your friend that you can stay in the water for 5 minutes. What is the probability that you survive the bet (no sharks will show up)?

c. If you go to the sea 5 times(30 minutes each time), what is the probability that you will observe 6 or more sharks in 3of the 5 times? (More than one distribution involved)

9. Natway, a national distribution company of home vacuum cleaners, recommends that its salespersons make only two calls per day, one in the morning and one in the afternoon. Twenty-five percent of the time a sales call will result in a sale and the profit from each sale is $125. Develop the probability distribution for sales during a five-day week. What is the mean and variance of this distribution? What is the expected weekly profit for a salesperson?

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#### Solution Preview

1. A coin is flipped twice.

a. What is the probability of two heads?

We know that P(head)=0.5. Since the two flips are independent, P(two heads)=P(1st flip is head and 2nd flip is head)=P(1st flip is head)*P(2nd flip is head)=0.5*0.5=0.25

b. What is the probability of a head in the second toss?

Using law of total probability, P(second toss is head)=P(second toss is head given first toss is head)*P(first toss is head)+ P(second toss is head given first toss is tail)*P(first toss is tail)=0.5*0.5+0.5*0.5=0.5

c. What is the probability of a head in the second toss given that a head was obtained in the first toss?

Using formula of conditional probability, P(second toss is head given first toss is head)=P(both toss are heads)/P(first toss is head)=0.5*0.5/0.5=0.5

d. At least one head in both flips.

We know that P(tail)=0.5 and both tosses are independent. Using complementary property of probability, P(at head one head)=1-P(no head in both tosses)=1-P(first toss is tail and second toss is tail)=1-P(first toss is tail)*P(second toss is tail)=1-0.5*0.5=0.75

2. The police placed three speed radars at three different locations (Palmyra, Dawson, and Slappey). The radars only work 40%, 30%, and 20% of the time. When you come to class late, the probabilities driving through the radar locations are 0.2, 0.3, and 0.5 respectively.

a. What is the probability of you getting a ticket?

Using law of total probability, P(getting a ticket)=0.4*0.2+0.3*0.3+0.2*0.5=0.27

b. Given that you got a ticket, what is the probability that you got it at Dawson?

Using formula of conditional probability, P(Dawson given getting a ticket)=P(Dawson and getting a ticket)/P(getting a ticket)=0.3*0.3/0.27=0.33

3. Imagine that you will have an interview for your dream job in Los Angeles. Since you do not want to miss the interview,you bought two tickets to LA. One withDelta and the other with US Airways. The probability of a Delta cancellation is 5% and the probability of a US Airways cancellation is 7%. Assuming that you ...

#### Solution Summary

The solution gives detailed steps on finding the distributions of the given data and calculating the probabilities using formula of relevant distributions. All formula and calculations are shown and explained.

Mean, mode, frequency distribution, normal distribution, Excel, scatter diagram, simple regression analysis, multiple regression analysis, regression analysis, linear programming

1 What is a mean? A median? A mode?

2 What is a frequency distribution?

3 What is a normal distribution?

4 What is the approximate probability that a data point will fall within one standard deviation of the mean of a normally distributed data set?

5 How does one begin to use Excel (or other comparable software) to calculate basic statistics on a set of input data?

6 How does one use Excel (or other comparable software) to calculate and display a scatter diagram?

7 What is linear regression? What is the difference between simple regression analysis and multiple regression analysis?

8 What is the most important concept in regression analysis?

9 What are the limitations of regression analysis?

10 What is linear programming? What are the two major types of linear programming problems?

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