T1

Coin side

Heads Tails

Count Percent Count Percent

10 tosses 5 50.00% 5 50.00%

20 tosses 11 55.00% 9 45.00%

30 tosses 17 56.70% 13 43.30%

If the coin is fair then the expected number of heads in 10, 20, and 30 tosses of coin are 5, 10, and 15 heads respectively. In the first 10 tosses, the observed number of heads is exactly the same as the expected number of heads. After tossing the coin 10 times, the number of heads is slightly higher than the expected number of heads. After tossing the coin another 10 times, the number of heads is still slightly higher than the expected number of heads. The cumulative percentage of heads varies a lot in the first few tosses of a coin. I hypothesize that if we continue tossing the coin then the cumulative percentage of heads stabilizes around 50%.

Comments:

Not necessarily. Over the long run, we would expect the percentages to be close to 50%. However, it doesn’t mean we would expect that in the short run.

Interestingly, the difference between the number of heads and number of tails tends to increase as the number of tosses increases. However, the percentages tend to approach 50%.

T3

Extended warranties are a cash cow for car dealers or financial institutions because, on the average, the total amount they receive as premiums is more than the total amount they pay out. Only a small fraction of customers will actually file a claim against their insurance company due to accidental damage or other unusual events. As an individual consumer it is a good idea to purchase an extended warranty when buying a new car to cushion the impact of unexpected expenses like expensive auto repairs. It is also peace of mind. During my last vehicle purchase, my wife and I decided to purchase a 100k mile extended warranty. We have used it one time in the last four years, but I still do not consider it a waste of money. I know that if something goes wrong with my vehicle I will be covered and not have an enormous repair bill that I have to come up with.

The estimated probability for the unusual events is used by the car dealers to determine the minimum premium they should charge. The higher the probability of filing a claim, the higher the premium they charge.

Comment:

This is important. While we know collectively the probability of cars needing repair is small, the problem for individual consumers is that we don’t know whether our vehicle will be the unlucky one.

You can use the overall low probability as the guideline to apply to your own vehicle, let’s say 10%. Now the issue boils down to how risk averse you are as an individual. Someone can tolerate this low risk, but some others cannot.

Subject | Mathematics |

Due By (Pacific Time) | 05/30/2014 12:00 am |

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