Homework #2

Homework #2

Due Tuesday February 1.

1. Play Chuck-a-Luck 1000 times (using Chuck.xls). How many times did you lose? How many times did you win $1? How many times did you win $2? How many times did you win $3? What was your total earnings or loss? What was the average earnings or loss per game?

2. Insure 1000 automobiles (using AutoInsur.xls), with a policy premium of $500. How much money does the inurance company make or lose total? How much money on average per car does the insurance company make or lose? Now try insuring 1000 automobiles with a policy premium of $1000 and answer the same questions.

3. You decide to open up your own casino. You invent your own game: The player rolls two 4-sided dice. If exactly one 4 appears, the player wins $1. If both dice are 4, the player wins $2. If no 4 appears, the player loses $1. How much on average will your casino make per game? Hint: Use the 4-by-4 diagram of problem 4 of homework #1 to find the probabilities of the three different possible outcomes (no 4, one 4, two 4s).

4. An insurance company writes a policy for a client who wishes to insure a machine valued at $300,000. It is determined that there is a 0.9 probability that there will be no loss (damage to the machine), there is a 0.05 probability of a $100,000 loss, and there is a 0.05 probability of a total loss. How much should the premium be on the policy so that the insurance company will not lose money (assuming no deductable)?

5. Construct a histogram for the data in the spreadsheet Unfair2.xls, which simulate the rolls of another unfair die. Construct a relative frequency histogram. Based on it, what are the probabilities of getting a 1, 2, 3, 4, 5 or 6?

6. Use the spreadsheet TenCoins.xls to simulate tossing 10 coins 1000 times. Create a histogram for the number of heads obtained for the tosses. (So there will be 11 columns; the first column will show how many times no heads appeared in 1000 tossses of the coins, the second column will show how many times one head appeared in 1000 tosses of the coins, etc.) Based on your results, what is the probability of getting exactly 3 heads in a toss of 10 coins? What is the probability of getting exactly 5 heads in a toss of 10 coins? [The probability distribution that controls how many heads are obtained when n coins are tossed is called the binomial distribution. See elementary statistics or probability texts for more information, including the simple formula for the probabilities. Binomial distributions are used to model the number of successes if a test is repeated many times.]

7. Create a spreadsheet that draws a relative frequency histogram for 500 random numbers. (Type =RAND( ) into a cell, then copy and paste into a range of 500 cells. Use the FREQUENCY function to count outcomes in the following categories: from 0.0 to 0.1, from 0.1 to 0.2, from 0.2 to 0.3; up the category from 0.9 to 1.0. See Notes 4.) The columns in your histogram should be roughly the same height since these numbers are supposed to be uniformly distributed. Based on this simulation, what is the probability of a number being between 0.0 and 0.1? Between 0.4 and 0.5?

8. Repeat exercise 3 but this time use the cell formula = -1.0*LN(RAND( )). This simulates an exponential distribution. It should be clear that some columns are taller than others. Based on this simulation, what is the probability of a number being between 0.0 and 0.1? Between 0.2 and 0.3? Between 0.2 and 0.5? Note, many numbers produced will be bigger than 1.0 now, so making the last column open-ended will count all the outcomes. [This exercise can be thought of as modeling the lengths of time between customer arrivals at a large store if one customer per minute arrives on average. If you wish to model the lengths of time between customer arrivals if, say 5 customers arrive per minute on average, replace the 1.0 in the cell formula by 0.2, the reciprocal of 5.]

1.	Play Chuck-a-Luck 1000 times (using Chuck.xls). How many times did you lose? How many times did you win $1? How many times did you win $2? How many times did you win $3? What was your total earnings or loss? What was the average earnings or loss per game?
2.	Insure 1000 automobiles (using AutoInsur.xls), with a policy premium of $500. How much money does the inurance company make or lose total? How much money on average per car does the insurance company make or lose? Now try insuring 1000 automobiles with a policy premium of $1000 and answer the same questions.
3.	You decide to open up your own casino. You invent your own game: The player rolls two 4-sided dice. If exactly one 4 appears, the player wins $1. If both dice are 4, the player wins $2. If no 4 appears, the player loses $1. How much on average will your casino make per game? Hint: Use the 4-by-4 diagram of problem 4 of homework #1 to find the probabilities of the three different possible outcomes (no 4, one 4, two 4s).
4.	An insurance company writes a policy for a client who wishes to insure a machine valued at $300,000. It is determined that there is a 0.9 probability that there will be no loss (damage to the machine), there is a 0.05 probability of a $100,000 loss, and there is a 0.05 probability of a total loss. How much should the premium be on the policy so that the insurance company will not lose money (assuming no deductable)?
5.	Construct a histogram for the data in the spreadsheet Unfair2.xls, which simulate the rolls of another unfair die. Construct a relative frequency histogram. Based on it, what are the probabilities of getting a 1, 2, 3, 4, 5 or 6?
6.	Use the spreadsheet TenCoins.xls to simulate tossing 10 coins 1000 times. Create a histogram for the number of heads obtained for the tosses. (So there will be 11 columns; the first column will show how many times no heads appeared in 1000 tossses of the coins, the second column will show how many times one head appeared in 1000 tosses of the coins, etc.) Based on your results, what is the probability of getting exactly 3 heads in a toss of 10 coins? What is the probability of getting exactly 5 heads in a toss of 10 coins? [The probability distribution that controls how many heads are obtained when n coins are tossed is called the binomial distribution. See elementary statistics or probability texts for more information, including the simple formula for the probabilities. Binomial distributions are used to model the number of successes if a test is repeated many times.]
7.	Create a spreadsheet that draws a relative frequency histogram for 500 random numbers. (Type =RAND( ) into a cell, then copy and paste into a range of 500 cells. Use the FREQUENCY function to count outcomes in the following categories: from 0.0 to 0.1, from 0.1 to 0.2, from 0.2 to 0.3; up the category from 0.9 to 1.0. See Notes 4.) The columns in your histogram should be roughly the same height since these numbers are supposed to be uniformly distributed. Based on this simulation, what is the probability of a number being between 0.0 and 0.1? Between 0.4 and 0.5?
8.	Repeat exercise 3 but this time use the cell formula = -1.0*LN(RAND( )). This simulates an exponential distribution. It should be clear that some columns are taller than others. Based on this simulation, what is the probability of a number being between 0.0 and 0.1? Between 0.2 and 0.3? Between 0.2 and 0.5? Note, many numbers produced will be bigger than 1.0 now, so making the last column open-ended will count all the outcomes. [This exercise can be thought of as modeling the lengths of time between customer arrivals at a large store if one customer per minute arrives on average. If you wish to model the lengths of time between customer arrivals if, say 5 customers arrive per minute on average, replace the 1.0 in the cell formula by 0.2, the reciprocal of 5.]