Betting on Lotteries?

My Toastmasters Talk (01.04.18) Don’t Bet on Lotteries

What’s today’s lottery number? I don’t know, and I don’t care, but many people do.

I suggest you ignore it, too. I'll tell you why.

Let’s take a three-digit lottery, # # #.

Possible values range from 000 to 999, a total of 1000.

If you guess right, you win the jackpot.

Most players will hit the jackpot  about 1 per 1000 times they play, unless lucky or cheating.

Your expected (average) reward each time is (1/1000) (jackpot).

If jackpot = $1000, over the long-run you’ll average $1 return per play,

After about a thousand plays, this is (1/1000) ($1000) (1000) = $1000.

You break even if it has cost you $1 a play.

Usually, the government taxes the winnings and the organizers deduct a profit, so you get less than jackpot, thus lose over the long run.

Some lotteries don’t have a fixed jackpot, but the winner(s) get a fraction of the total bet, after taxes and profits are taken out first. If there are N winners, each gets jackpot/N. What number to bet? All are equally likely to result, but you don’t want to pick a popular number, or you may have to share any winnings with more people. For example, if I choose my telephone area code, 845, there is a good chance that some others will do so, too, and if 845 comes up, many will split the pot. If it’s a four-digit lottery, and I choose my birthday month and date, 1221, I will likely share any winnings with other December 21^st babies. Other “popular” choices to avoid are 123 234 345 and so on, 111 222 333 and so on, again numbers that others might likely choose. Find a way to get a random number, and if that looks too “popular,” choose another?

What about Mega Millions and other national lotteries?

Recently, Mega Millions had a jackpot of $343 million. The odds of getting the right six numbers is 302 million, so the long-term average is $343/302 or $1.14, but each ticket costs $2, so on the average, bettors lose almost half.

The Mega Millions lottery also has prizes for getting fewer than six balls correct, so the odds are somewhat better than this.

UTILITY

Rather than measuring the outcomes in dollar terms, we would like to measure the value to the participant, the utility. $1000 may or may not be 1000 times more valuable to player A than $1. Player B may have a different valuation from player A, changing the calculation yet again.

RISK-AVERSE AND RISK-PRONE

Further some people feel loss more deeply than others and gain less intensely than others, making them risk-averse. Their opposites are people who are less sensitive to loss and more sensitive to gain. “Know thyself,” said Socrates. 

Your reward/loss sensitivity should come into consideration.

KNOW THE ODDS: BIRTHDAYS

What is the probability your birthday day of the month is an odd number? Is it 50%? My date is the 21^st, odd. First guess is that half the days are odd and half even. Ignoring leap years, for 365 days, that can’t be exactly right. Moreover, there are four months with 30 days, and one with 28 days (half odd and half even), but the seven months with 31 days have an extra odd day. So, there are 11*15+14+7 = 186 odd days and 11*15 +14= 179 even days, so your probability of being born on an even day is 179/365=0.490 and on an odd day, 186/365=0.510. the ratio is 186/179= 1.039, a 4% difference.

What is the probability you were born on the 15^th? Of 365 days, 12 are on the 15^th, so the probability is 12/365= 0.033, a bit over 1/30, about 3%.

What is the probability you were born on the 31^st? Only 7 months have 31 days, so you probability is 7/365=0.019, about 2%.                                                                                                                                         

What is the probability that in a group of N people, 2 or more have the same date of the month for their birthday? Much higher than you expect, as N gets large. Try it among a group, going one by one. Even among 10 people. it is quite likely. Each new date gets compared against the dates already mentioned. Roughly, you have cumulative probabilities 0/30 + 1/30 + 2/30…. [There is a more exact formula available.]

So, knowing the odds can be difficult.

ON THE OTHER HAND

“It’s not whether you win or lose, but how you play the game,” sports writer Grantland Rice advised. Sometimes we say that the journey is more important than reaching the destination. Betting can be entertaining, at least. 

If you can treat betting as entertainment and the losses as entertainment expenses, and if you get your money’s worth by imagining winning while awaiting the outcome, and you don't mind losing, then betting on a lottery is rational…just don’t overdo it, don’t bet more than you can afford to lose!