MEGA Millions Lottery

The chance of winning the MEGA Millions lottery jackpot is 1 in 303 million (more precisely, 1 in 302,575,350.)  To put this in perspective:

If you buy 50 tickets per week you will win the jackpot once every 116,000 years.

If every time you drive a mile you buy a MEGA Millions ticket, you will make an average of 633 round trips to the moon before you win the jackpot.

Suppose you have one friend in Canada.  Suppose you put the names of everyone in Canada in a hat and draw a name at random.  Suppose you then buy a MEGA Millions ticket. You are 9 times more likely to draw your friend’s name than you are to win the jackpot.

Since the chance is so low of winning the Mega Millions jackpot, why are there winners?   It’s because of the “Law of Very Large Numbers”:  Given enough opportunity, any weird thing will happen just due to chance.   In other words, as long as there is opportunity for weird things to happen, chance alone explains strange coincidences, mystical occurrences, and other bizarre events.  In the case of the MEGA Millions lottery, there are jackpot winners because so many people buy tickets.  The chance may be almost zero that you or I will win the jackpot, but if more than 300 million tickets are sold, there is a good chance that someone will win.

Luck is a group phenomenon.  In the case of the Mega Millions jackpot, for every lucky winner, there is, on average, an unlucky group of 303 million losers.

When you buy a Mega Millions ticket, you pick Pick 5 numbers from 1 to 70 and 1 MEGA number from 1 to 25.   There are about 303 million ticket combinations.  Since the winning ticket is chosen at random, that’s why the chance of winning the MEGA Millions jackpot is about 1 in 303 million.

If you were to buy every possible ticket combination, it would cost you about $303 million.  When nobody wins for awhile, since prize money carries over to the next drawing when there is no winner, sometimes, the jackpot might  exceed $303 million.  Suppose the jackpot reaches $500 million.  Then if you invest $303 million in buying every possible ticket combination, you are guaranteed to win the jackpot, giving you an apparent profit of $500 million minus $303 million, or $197 million.  Unfortunately, if you fill out 3 tickets per minute, 24 hours per day, it will take 192 years to fill out all possible ticket combinations.  And even though you would be assured of winning the jackpot, if some other lucky player also wins, you would need to split the pot with them.  Also, you need to pay taxes on net gambling winnings.  So making a large bet like this is probably not a good idea, even if you figure out a way to do it.




Roulette and the Double-Down Strategy

Quote from a Las Vegas gambler:  “I hope I break even this week.  I need the money.”

A roulette wheel is divided into 38 sections, numbered from 1 to 36, 0 and 00.  18 of the sections numbered from 1 to 36 are black and 18 are red.  The sections 0 and 00 are green.

You can bet on individual numbers, combinations of  numbers, or colors, before the wheel is spun, by placing chips in appropriate sections on the betting layout

The wheel is spun by a casino employee, who then spins a ball along the wheel in the opposite direction.  The ball comes to rest in one of the 38 sections, which then becomes the winning section.  Players who bet on the winning section are paid off accordingly.   For example, a winning bet on #17 pays 35 to 1 odds.  A winning bet on red sections pays 1 to 1 odds, or “even money.”

What happens to the roulette gambler in repeated play?

Since the chance is 18 in 38 that the winning section will be red, the “law of averages” states that in repeated play red will come up an average of 18 times in 38 spins.  Similarly, #17 will come up, on average, once in 38 spins.  So if you repeatedly bet $1 on red, on average, you will win 18 times and lose 20 times in every 38 bets, for an average net loss of $2 per 38 spins = $2/38 = $.053 (5.3 cents) per bet.  Likewise, since the chance is 1 in 38 that #17 will be a winning section, the law of averages states that in repeated play, #17 will come up about once every 38 spins.  So if you repeatedly bet $1 on #17, on the average you will win once and lose 37 times in every 38 bets, for an average net loss (taking into account the payoff odds) of 35x$1 – 1x$37 per 38 spins, or $2/38 = $.053 per bet.

For bets like this, the player will eventually lose at the rate of 5.3% of all money bet and casino will make a 5.3% profit.

Are there any strategies that circumvent the casino’s 5.3% profit margin (sometimes called the “House Edge”).  Consider the “double-down” strategy:

  • On the first bet, wager $1 on red.  If red comes up, you win $1. Quit.
  • On the 2nd bet (if red didn’t come up on the first bet): Double your bet and bet $2 on red. If red comes up, you win $2, covering your $1 loss on the first bet and leaving you a $1 profit. Quit.
  • On the 3rd bet (if red didn’t come up on the first two bets): Double your bet and bet $4 on red. If red comes up, you win $4, covering your previous $1 and $2 losses and leaving you a $1 profit. Quit.
  • Etc

By the laws of chance, eventually red has to come up, at which point you quit a winner!!!

Is there anything wrong with this strategy?


All casino games have a house limit. If you encounter an unlucky streak of losses, the amount you need to bet may exceed this limit, thus causing you to not cover your losses.

Most people have a limit. If you encounter an unlucky streak of losses, the amount you need to bet may exceed this limit, also causing you to not cover your losses.

Although unlikely, if red fails to come up 15 times in a row, on the 16th bet, you must wager $32,768 in an attempt to come out $1 ahead.  Most casinos will not allow such a bet.

Alas, it turns out that the double-down strategy, although deceptively appealing, is no different from other roulette bets:  In the long run, the gambler will still lose at the rate of $.053 per dollar bet.

It should be noted that the double-down strategy says to quit as soon as you win. What does it mean to quit? Does it mean that as soon as you win your dollar you never come back to the roulette table again?  Or does it mean to go have a drink and then start over?  For most gamblers, it means the latter.  Sadly, if you quit forever, you wouldn’t be a gambler anymore.