Misunderstanding the Probability of Winning
Thursday, November 29, 2012
Charles Rotblut, CFA
AAII Journal Editor

AAII Resources

Asset Allocation and Retirement
Your retirement portfolio should reflect your time horizon and tolerance for risk.

Explaining the Value Tilt
Value stocks have outperformed growth stocks historically.

AAII Discussion Boards
Do you play the lottery, and how do you pick numbers?

Most Popular AAII Articles

  1. “U.S. Fiscal Uncertainty and Your Portfolio”
  2. “Making Sense of Master Limited Partnership Tax Rules”
  3. “Aging and Investing: The Risk of Cognitive Impairment”


Sentiment Survey

This week’s AAII Sentiment Survey results:
  Bullish: 40.9%, up 5.1 points
  Neutral: 24.7%, up 1.3 points
  Bearish: 34.4%, down 6.4 points

Long-term averages:
  Bullish: 39.0%
  Neutral: 30.5%
  Bearish: 30.5%

Take the AAII Sentiment Survey »


A common behavioral error occurred this week: Many people thought they could increase their odds of winning the $587.5 million Powerball jackpot by purchasing more than one ticket. On the surface, the logic makes sense. Buy two tickets instead of one and you double your odds. Buy 50 instead of one, and your odds are 50 times better. The problem with such logic is that it doesn’t consider whether buying the extra tickets has any significant impact on the probability of winning.

Powerball, like other forms of gambling, has a fixed number of outcomes. The lottery game picks five unique numbers between 1 and 59. A sixth number, the “Powerball,” is then drawn. The Powerball number ranges between 1 and 35. A total of 175,223,510 combinations can be formed. Since there are a fixed number of combinations, it is easy to calculate the probability of winning the jackpot for any number of tickets purchased. We simply need to divide the number of tickets purchased (assuming each has a different combination of numbers) by 175,223,510.

If you bought one ticket, you had a 0.00000057% chance of winning. Not very good, but a ticketholder in Missouri and a ticketholder in Arizona did win last night. Buying two tickets increased your odds to 0.00000114%. Yes, this was technically twice as good, but your odds were still very low. Splurged and bought 100 tickets (a $200 expenditure)? Your probability of winning only improved to 0.00005707%.

If your goal was to just to have a 1% chance of winning, you would have had to spend $3,504,470, at a price of $2 per ticket. (You would also need several very patient store clerks, the free time to have all of those tickets printed, a system to avoid any duplicate tickets being selected and a method for checking all of those tickets.) Even with the large expenditure, there was a 99% chance you wouldn’t have won the jackpot. You also didn’t have any guarantee of winning enough of the smaller prizes to compensate for the money you spent on tickets.

I bring this up because part of both gambling and investing is understanding the probabilities of winning (making) or losing money. In a game with known fixed odds (e.g., the lottery, poker, roulette, etc.) you can assess how risky a bet is with math. When it comes to investing, the outcome is not always finite (a stock can theoretically keep appreciating in price), but you can still assess the risk by considering what has worked historically. Over the long term, we know value stocks perform better than growth stocks and small-cap stocks beat out large-cap stocks. Companies that initiate or raise their dividends deliver higher total returns than those that don’t pay a dividend. Investment-grade bonds are less likely to default than junk bonds. Funds with lower fees have to beat their benchmarks by a lower margin than funds with high fees to give shareholders the same amount of profit. Diversification and a long-term view will help your portfolio more than a concentration in a few investments and a short-term view.

Keep in mind that, by definition, probability is not the same as certainty. You could do everything right with your portfolio and still not be happy with your returns. Similarly, you could ignore the statistics above, spend $250 on Powerball tickets and win the jackpot. But, if you stop to consider the probabilities of making or losing money, the odds of you making better financial decisions (and not overspending on lottery games) are likely to improve.

More on AAII.com

Because our focus is investing, we don’t have any articles about gambling or the lottery. What we do have are articles that can help make you a better investor, thereby increasing your investing “luck.” Two of these articles are listed below.

The Week Ahead

Just three S&P 500 member companies are scheduled to report earnings next week: AutoZone (AZO) on Tuesday, SAIC (SAI) on Wednesday and H&R Block (HRB) on Thursday.

The week’s first economic reports will be the November ISM manufacturing index and October construction spending, both of which will be released on Monday. Tuesday will feature November motor vehicle sales. The November ADP employment report, revised third-quarter productivity, October factory orders and the November ISM non-manufacturing report will be published on Wednesday. November jobs data, including the unemployment rate and the change in nonfarm payrolls, plus the preliminary December University of Michigan consumer confidence survey will be released on Friday.

St. Louis Federal Reserve Bank President James Bullard will speak on Monday.

AAII Sentiment Survey

Bullish sentiment registered above 40% for the first time since August 23, 2012 in the latest AAII Sentiment Survey. Bearish sentiment continues to stay above its historical average, however.

Bullish sentiment, expectations that stock prices will rise over the next six months, rose 5.1 percentage points to 40.9%. This reading ends a 13-week streak of optimism coming in below its historical average of 39.0%.

Neutral sentiment, expectations that stock prices will stay essentially unchanged over the next six months, edged up 1.3 percentage points to 24.7%. Even with the increase, this is the 10th time in the past 11 weeks that neutral sentiment is below 30%. The historical average is 30.5%.

Bearish sentiment, expectations that stock prices will fall over the next six months, fell 6.4 percentage points to 34.4%. This is an eight-week low. Even with the drop, pessimism is above its historical average of 30.5% for the 14th consecutive week and the 30th out of the last 34 weeks.

More individual investors are describing themselves as bullish than bearish for just the second time in the past 10 weeks. The bull-bear spread, which measures the difference between bullish and bearish sentiment, is also at its most positive level since August 23, 2012. The current bull-bear spread is 6.5.

Though this week’s survey signals an increase in the level of optimism, it is important to note that this is only the second time since March 29, 2012, that bullish sentiment is above its historical average of 39%. Individual investors remain cautious and failure by Congress and the president to avoid the fiscal cliff would likely have a damaging impact on individual investors’ moods. Though some AAII members are encouraged by signs of continued economic growth and the preliminary holiday shopping data, many remain concerned about the pace of economic growth, ongoing political gridlock and Europe's sovereign debt crisis.

This week’s special question asked AAII members what this year’s holiday shopping trends are saying about the economy. Responses were mixed, with the largest group of respondents saying the early data shows signs of an improving economy, or at least better consumer sentiment. Several members thought consumers are either ignoring the macro environment (including the possibility of the fiscal cliff fears) or are just tired of not spending. Some respondents are worried that consumers are spending money they do not have. There were also several who thought the initial data did not provide much insight into the overall health of the economy.

Here is a sampling of the responses:

  • “With holiday spending up slightly, consumers are not letting the fiscal cliff threat spoil their holidays.”
  • “I’m encouraged by the volume of shoppers reported by the media, but I hope that folks are not spending themselves back into difficult situations.”
  • “Consumers have become more willing to spend.”
  • “Consumers don’t have much "extra" money, so they are shopping for the lowest priced products.”
  • “It is very hard to tell as of now, but aside from Black Friday, I feel that people are being careful with their money.”
  • “Black Friday does not a shopping season make.”
The historical average for neutral sentiment was adjusted down by a half a percentage point this week, from 31% to 30.5%. The historical average for bearish sentiment was revised up, from 30% to 30.5%. These revisions reflect a trend we have seen develop in the weekly readings. Over time, we make additional adjustments to the historical averages as the data warrants.

» Take the sentiment survey