**Due Date: Tuesday, March 17, 2015**

**Question 1****:**

A gambler plays a sequence of games that she either wins or loses. The outcomes of the games are independent, and the probability that the gambler wins is 2/3. The gambler stops playing as soon as she either has won a total of 2 games or has lost a total of 3 games. Let *T*be the number of games played by the gambler.

- Find the probability mass function of
*T*. - Find the mean of T.
- Find the standard deviation of
*T*.

**Question 2****:**

According to the current Commissioners' Standard Ordinary mortality table, adopted by state insurance regulators in December 2002, a 25-year-old man has these probabilities of dying during the next five years:

Age at death | 25 | 26 | 27 | 28 | 29 |

Probability | 0.00039 | 0.00044 | 0.00051 | 0.00057 | 0.00060 |

(i) What is the probability that the man does not die in the next five years?

(ii) An online insurance site offers a term insurance policy that will pay $100,000 if a 25-year-old man dies within the next 5 years. The cost is $175 per year. So the insurance company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Its loss depends on how many premiums were paid, as follows:

Age at death | 25 | 26 | 27 | 28 | 29 |

Loss | $99,825 | $99,650 | $99,475 | $99,300 | $99,125 |

What is the insurance company's mean cash intake from such polices? Hint: These losses should be considered as negative numbers whereas the payments are positive.

(iii) What annual premium should the insurance company charge a 25-year old for a term life insurance policy if it is to expect a positive profit.

**Question 3****: (Experimental probability)**

In a marketing ploy to boost sales, a cereal manufacturer puts pictures of famous athletes in boxes of cereal. The manufacturer advertises that 40% of the boxes contain a picture of Peyton Manning (an NFL superstar), 30% a picture Usain Bolt (a track superstar), and 30% a picture of Abby Wambach (a soccer star). We are interested in the random variable defined as the number of cereal boxes a person needs to buy in order to get the complete set of pictures?

- a) Use Table B to simulate the random process of buying cereal boxes until all three pictures are collected. The resulting number of boxes is a simulated observation of

. Specifically, here is how this is done. Since the digits from 0 to 9 are equally likely, assign {0,1,2,3} to Manning, {4,5,6} to Bolt, and {7,8,9} to Wambach. Pick any line in table B and obtain the minimal sequence of single digits that will give you all three cards as represented by the assignment above. The length of the sequence is an observed value of . For example, using line 101, the sequence 19223950340… gives us (do you see why?)

- b) Here comes the tedious part. Repeat (a) 50 times. Make sure you change location (line number) in table B each time when you are generating a new observation.
- c) Make a table (such as the one below) of the approximate distribution of based on the 50 observations you obtained in (b).

x | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Etc… |

Count | ||||||||

p(x) |

- d) Obtain a histogram for the approximate distribution of and describe the overall pattern. Do it by hand.
- e) Find the approximate probability that a person will have to buy 6 or less boxes to get the complete set of pictures.
- f) Find the approximate mean value of .

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