Choices

Probabilities and Averages

Probabilities are often used to calculate the expectedaverage outcomes for a decision

For example, a book publisher may know that new mystery novels by unknown authors most often lose money, but sometimes the reviews and word-of-mouth are strong and large sales develop.In the latter case, there will usually be a series of books by that author (who now develops a following) and the publisher makes a lot of money.Over the long run, the publisher can make money based on the average of many moderate losses and a few large gains. For example, suppose the publisher knows that 90% of such first novels fail to break even, and the loss in such cases averages about $40,000 (includes advance to the author, editing and typesetting, production of an initial press run, and marketing costs, offset by some modest sales). In about 8% of all cases, the novel is fairly successful and leads to additional successes; net profit in such cases will average about $200,000 (spread over several years). Finally, in about 2% of all cases, there is a huge success, with a profit averaging about $2 million over several years. The graph here shows the information:

One can calculate the average result by considering what will happen in such 100 novels, with 90, 8, and 2 falling in these three types. Across 100 such first novels, the publisher expects to lose 90x$40000 = $3,600,000 on the failures; to gain 8 x $200000 = $1,600,000 on the fairly successful novels; and to gain about $4 million on the two huge successes. Taken together, this gives a net gain of $2 million for these 100 novels; this means an average of $20,000 net profit per attempt of this sort. A short form of calculation for this average simply multiplies the different gains (or losses) by the respective probabilities and adds up these products:
.90(-40000) + .08(+200000) + (.02)(+2000000) =20000.

Since all costs are already accounted for, it seems like a good decision to publish each such novel -- on the average, you expect to gain $20,000 profit for each such case. The calculation of average return, using probabilities, can be a good way to make decisions (to publish or not (correctly indicate relative frequencies of the different sorts of events, and provided that one does not assume unreasonable risks. For example, under the above assumptions, the huge successes might not happen quite on schedule, so the publisher needs "deep pockets" to protect against being wiped out by a long series of small failures.

"Small" vs. Catastrophic Risks
 
In the publishing example, as outlined, there was no risk of a single catastrophic failure.  One could imagine a publishing scenario in which catastrophe is possible, eg  a libel suit could bankrupt a publisher.  Where possible, it is wise to protect against catastrophe by purchasing insurance.  One reduces the average profit somewhat, by paying the insurance premium, in order to avoid the risk of large losses.  For the insurance company, with its very large financial base, the same large loss is not a catastrophe, and so they make a profit by averaging together many insurance policies issued, only a few of which make large claims.
 
 To pursue this last example, suppose that one out of every 5000 novels produces a libel suit, and that the average legal and settlement costs when a publisher is sued are about $600,000.  If the insurance company charged publishers an insurance premium of $250 per novel published, then across a representative set of 10000 novels, they would  $2,500,000 in premiums and pay out $1,200,000 in costs, making a gross profit (prior to expenses and commissions) of $1,300,000, or $130 per novel.

This is shown by the following graphs

Here is another example, looking at insurance from the standpoint of the consumer.  A friend of ours used to purchase insurance for all packages that she sent through the mail.  In those days, insurance on a package with value $20 cost about $.60; the probability that such a package would be lost in the mails and never delivered was about 1 in 250 (0.4%, or .004).  Thus, if our friend sent 1000 packages in her lifetime, each valued at $20, she would expect to lose 4 of them, for a total cost of $80.  The insurance for 1000 packages cost $600, however.  Thus, she  would have been far better off, over a lifetime, spending $80 to replace the 4 lost items than spending $600 total on insurance.  On the average, she was losing $.52 per $20 package mailed, by buying insurance, as opposed to losing $.08 on the average, per $20 package mailed, by not buying the insurance.  Here again, there is an implicit argument involving tolerable risk.  My friend could afford to replace a $20 purchase that was lost.  It would be unpleasant, but not catastrophic.  On the other hand, if a single package cost $20,000, rather than $20, with insurance costing $600,rather than $.60, my friend might decide that losing the $20,000 would be not merely unpleasant but ruinous.  Paying the premium of $600 would be highly unpleasant, but not ruinous.  So she should buy the insurance.  On the average, in the long run, she is still losing by purchasing such insurance, but this is a case where the long run average doesn't matter much.  She is not going to be sending a  $20,000 package very often.  She can somehow manage a handful of $600 premiums over a lifetime, but she cannot manage a loss of $20,000 even one time.  A large corporation that sends out $20,000 packages often is better off avoiding  insurance -- it can better afford to replace 4 out of every 1000, at a total cost of $80,000, then to insure all 1000 at such rates, with total cost of $600,000.

Time Perspective
 
Note that these examples all operate on a long time perspective.  In the immediate situation, it may look OK to pay just $.60 to insure a package, and it may have made my friend feel reassured to know that she would be reimbursed if it was lost.  By looking at this policy in the perspective of 1000 packages over a period of years, she could recognize that she would expect to lose $520 by purcha whether the feeling of reassurance each time she mailed a package is worth $520 paid out over a long period of time. Perhaps she would decide it was worth it, or perhaps she would decide that occasional $20 losses are no big deal and the insurance is not worth the cost.