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Probability 101
By JLP | April 11, 2008
I’m in the process of reading Peter Bevelin’s awesome book, Seeking Wisdom - From Darwin to Munger (Not an Affiliate Link). I HIGHLY recommend this book for anyone interested in investing and behavioral finance. As boring as that sounds, this book is a page-turner.
One of the sections of the book that I found most interesting was this illustration of probability on page 151:
A lottery has 100 tickets. Each ticket costs $10. The cash prize is $500. Is it worthwhile for Mary to buy a lottery ticket?
The expected value of this game is the probability of winning (1 in 100) mulitiplied with the prize ($500) less the probability of losing (99 out of 100) mulitplied with the cost of playing ($10). For each outcome we take the probability and multiply the consequence (a reward or a cost) and then add the figures. This means that Mary’s expected value of buying a lottery ticket is a loss of about $5 (0.01 × $500 - 0.99 × $10).
He goes on to say…
Mary has a 1% chance of winning the lottery and if she wins, her gain is $490. She has a 99% chance of losing $10.
What happens if Mary buys 10 tickets?
If Mary were to buy 10 tickets, she would have a 10% chance of winning and her gain would be $400. She has a 90% chance of losing $100.
I don’t think I would play this lottery!
This is a very simple example but interesting nonetheless.
I always wondered about this stuff but never gave it much thought. I actually hated stats class in college. I had to take three semesters of stats because the one semester I took in college in Kansas wouldn’t transfer to my college in Texas. To top it off, the Texas college required TWO semesters of stats. Lovely! I just wish I would have paid more attention in class!
It’s not my intention to turn this into a statistical blog, but I do hope to explore this topic some more in the future.
Topics: Financial Math Basics |
April 11th, 2008 at 1:21 am
Here’s my reason for not playing this lottery:
Let’s say Mary buys 50 tickets at $10 each, for a $50 “investment”. At this point, if she wins, her net gain is zero. Yet, she only has a 1 in 2 chance of winning.
In short, buying half the tickets gives you a 1 in 2 chance of breaking even!
Thanks but no thanks, lol
April 11th, 2008 at 7:04 am
With the probability now known , I don’t think I would buy lottery .
great article , thanks,
Tracy Ho
wisdomgettingloaded
April 11th, 2008 at 8:59 am
Unfortunately, probability doesn’t always translate directly into real-life situations.
Let’s take your example of the lottery, except we’ll change things up a little.
Mary is 50 years old and approaching retirement. She’s been financially savvy for her entire life and has accumulated $1M in cash.
Donald Trump decides to hold a lottery for only Mary. One ticket costs $1M, and she has a 50% chance of winning $10M.
If you looked at just probability, her EV is -(0.5 x $1M) + (0.5 X $10M), or +$4.5M. Does that mean she should buy the ticket? Obviously, no.
State lotteries have been accused of being “taxes on the poor.” If you take the inverse of my example, I think you can understand why that statement holds a lot of truth.
April 11th, 2008 at 9:26 am
I loved statistics in college. The professors were always boring, the concepts highly graspable (if you put in enough time), and the students uninterested - all of which combine to make for an easy A for the motivated individual.
If people were rational, they’d never play the lottery. Thank God for most state governments that most people are not rational.
April 11th, 2008 at 10:15 am
I like to think of probability in terms of coin flips; ie, how many times do I have to flip a coin and have it land on heads in a row to match the probability of something.
In my blog, I determined that to win Megabucks, I’d need to get 22 heads in a row. I love this stuff.
More here:
http://stupidmoneyhacks.com/?p=54
Mr. Stupid
April 11th, 2008 at 10:46 am
I really enjoy statistics, and especially find applications to gambling interesting.
In regards to the lottery, I usually buy a ticket just for fun when the Powerball is above 200 million or so. I believe that above around 250 million, there is actually a positive expectation for an individual player. The odds of winning are something like 1 in 140 million, and the cash value would be a little above that. Granted, millions of other people buy tickets as well, the prize pool will be diluted, especially if someone else wins the jackpot, but it’s still interesting to think about. When the prize has gotten in the 300 million range, I’ve often wondered if there are any billionaires out there who have considered buying all 140 million possible tickets, in order to gain a quick return of 50-100% on their money. My guess is probably not, since there would be too much risk of sharing the jackpot, and you could be out a lot of money, especially considering taxes and what not, not to mention the logistical issues of buying that many tickets!
Anyway, the lottery you discuss is indeed a terrible deal. It would be like betting on a coin flip and only getting paid half of what you bet when you win.
April 11th, 2008 at 12:01 pm
actually it is:
0.01 × $490 - 0.99 × $10
The cost of a ticket must be counted for the win and the loss.
April 11th, 2008 at 12:08 pm
Again with the lottery stuff. I sometimes buy a ticket, but I know that the probabilities are literally astronomical. I heard somewhere that your chances of being hit in the head by a meteor are greater than those of winning the lottery jackpot. I cannot track down a source on that one though, so take with a grain of salt.
April 12th, 2008 at 5:40 pm
Joewatch’s point is exactly on the money:
1. You should understand basic probability because it is so important in life, BUT
2. You should first make the Life Decision then look at the odds …
… I tried a slightly different take on this using Deal or No Deal as an example (hope you don’t mind the link … it’s relevant):
http://7million7years.com/2008/02/08/deal-or-no-deal/
AJC.
April 13th, 2008 at 5:58 am
I’m not an economist or a mathematician, but for me I think the concept of marginal utility is relevant to why I sometimes buy a lottery ticket. Now, most of the time I follow the simple, rational/mathematical analysis outlined above, so I don’t buy lottery tickets. But now and then, usually when the Powerball prize gets very large, I’ll go out and buy one ticket even though the mathematical expectation is that I an extremely likely to lose a small amount of money (ca $1) by doing so. Here is where the marginal utility concept comes in. That $1 loss, especially when it only happens once or twice a year, is pretty much invisible to my finances. However, a few hundred million bucks would be life-changing. So even if the chance of winning is infinitesimal, I sometimes play when there is a big payout.
The final reason I think I play is that it’s just a little bit entertaining to briefly think I have a shot at winning. There’s just the tiniest whiff of adrenaline when you check your ticket — not bad for a buck.