Should you play the lottery? How can you increase the chances of winning?

Posted by Andrew Price on

fired and glazed ceramic jar with "Lottery Money" dark lettering on white background

Many people of all ages enjoy playing the lottery, and indeed for a very long time. The earliest recorded state lottery dates back to the Han Dynasty in China some time between 205 and 187 BC, possibly to finance the building of the Great Wall of China, according to Wikipedia.  Similar to the state lotteries of Ancient China and unlike gambling at a modern day American casino, the money raised from the state lottery is usually given to state and federal governments to be disbursed for public use. In fact, the California Lottery states that “95 cents of every dollar you spend on Lottery games goes back to the community through contributions to public schools and colleges, prizes and retail compensation.”

In California, the disbursement of funds is controlled by the State Controllers office. The state requires that the Lottery must disburse:

  • no less than 50% of revenue to prize payouts
  • no more than 13% of revenue to Administrative expenses
  • Prize payouts and educations funds must not be less than 87% of sales
  • 100% of any other income must go towards education

 For the fiscal year of 2020-2021, the California Lottery generated more than $1.88 billion for public education, marking the twenty first consecutive year of more than $1 billion being transferred to public education from the California Lottery. For the last 12 months ending June 30th, 2020, administrative expenses totaled only 11.98% of revenue, or $793,133,391. 

Over the years California Lottery funds have been distributed to K-12 schools, community colleges, the California State University, the University of California, the Hastings College of the Law, the Department of Corrections & Rehabilitation-Division of Juvenile Justice, the Department of Education-State Special Schools, the Department of Developmental Services, and the Department of State Hospitals. However, California Lottery funds account for less than 1.5% of all education funding in the state. By this estimate, Public education in California costs tax payers and the state treasury over $125 billion each year.

But, although the lottery aims to provide a public service, do the ends justify the means? According to California Council of Problem Gambling, approximately 3.7% of the population are problem gamblers. Based on California’s population of 39,538,223 in 2020, it stands to reason about 1,462,914 people have a gambling problem. It could be even higher in California, when considering states like Utah and Hawaii ban gambling outright.

Okay, but you live in one of the many states that allow gambling, should you play your state’s lottery? Gambling institutions must provide the odds of winning each game they publish. Let’s take California Superlotto Plus, for example. Of the 3 high value games available, it has the best odds of you winning the grand prize at 1 in 41,416,353 per draw. Though these odds don’t sound good, the odds of one ticket winning the Mega Millions is 1 in 302,575,350 and the odds of one ticket winning the Powerball are 1 in 292,201,338. Basically, if every person in the state of California purchased one lottery ticket (and no duplicates) one person would win the California Superlotto Plus each drawing. Whereas, almost every person in the United States (population estimate of 334,375,357 on February 12, 2023) would have to purchase a lottery ticket each drawing for there to be a winner every drawing.

Now you know the odds, but how are they calculated? Odds are calculated based on the game rules. For the lager games using balls drawn from a globe, we can use the statistical equation for combinations to calculate the odds. The statistical combinations equation applies only when the order of the numbers do not matter. If order matters, then you need to use the statistical permutations equation. Since order does not matter for Superlotto Plus, we can use the formula:

 𝑛𝐶𝑟 = 𝑛! / (𝑛𝑟)! 𝑟!


The “!” in this equation stands for factorial. A factorial is a shorthand method of demonstrating the multiplication of sequential numbers from the starting number down to one. For example, 5! is the same as 5 x 4 x 3 x 2 x 1 = 120. The n stands for the number of options, in our case, balls numbered 1 to 47. The r stands for the number of balls picked, in our case, 5 balls are picked.

The current rules of Superlotto Plus state you need to pick five numbers between 1 and 47 and one Mega number between 1 and 27. Since the order by which the numbers appear does not matter, this means we can use the combinations formula to calculate the odds of winning the grand prize:


𝑛𝐶𝑟 = [𝑛! / (𝑛𝑟)! 𝑟!] x      [𝑛! / (𝑛𝑟)! 𝑟!]

𝑛𝐶𝑟 = [47! / (47-5)! 5!] x      [27! / (27-1)! 1!]

𝑛𝐶𝑟 = [47! / (42!)(5!)] x     [27! / (26!)(1!)]


𝑛𝐶𝑟 = 

[(47 x 46 x 45 x 44 x 43 x 42!) /   x       [27 x 26! /

(42!)(5 x 4 x 3 x 2 x 1) ]                     26! 1!]


𝑛𝐶𝑟 = [47 x 46 x 45 x 44 x 43 /    x       [27/

             120]                                            1]


𝑛𝐶𝑟 = [184,072,680 / x.    [27]


𝑛𝐶𝑟 = [1,533,939] x [27]


𝑛𝐶𝑟 = 41,416,353


So now you know your odds of winning the big prize, but another thing to consider is the payout for that prize. Let’s take their listed odds of winning the various prizes into consideration for the previous game played on February 11, 2023 and compare.


Matching Numbers

Odds 1 in

All 5 of 5 and Mega


All 5 of 5


Any 4 of 5 and Mega


Any 4 of 5


Any 3 of 5 and Mega


Any 3 of 5


Any 2 of 5 and Mega


Any 1 of 5 and Mega


None of 5, only Mega



Matching Numbers

Winning Tickets

Prize Amounts

5 + Mega






4 + Mega






3 + Mega






2 + Mega



1 + Mega







If each ticket cost $1 each, at what point would you break even?

Assuming no duplicate tickets, the jackpot would need to be above $41 million (and you would need to purchase 41 million tickets to guarantee a winner would be drawn. However, what if someone else has the same strategy? Then you would need to split the winnings with them, bringing your winnings down to $20.5 million. There’s also federal income taxes that need to be paid on your winnings. California does not withhold or tax California Lottery winnings, however your winnings may put you in a higher federal tax bracket and you’ll be obligated to pay that to the IRS.

Let’s assume you are the only person playing the lottery, now how big does the pot need to be for you to guarantee a win and breakeven?


Federal tax rate capped at 37% for married, 35% for single.

x = $41,416,353/(1-0.37)


x =  $41,416,353 / 0.63



x = $65,740,242.86



You would need the pot to be nearly $66 million and you would need to purchase about 41.42 million tickets (no duplicates) in order to ensure a win and break-even after taxes.


x = $65,740,242.86 - ($65,740,242.86*0.37)


x = $65,740,242.86 - $24,323,889.86



x = $41,416,353


But what about the other prizes?


In order to calculate the odds of winning the lesser prizes we need to add in an additional equation and divide our total odds by that number. For example, what are the odds of winning the prize if your ticket reads 4 out of 5 of the numbers and you get the Mega number correct?


The variables we will use are:

n = the total number of options/balls

r = the number drawn

k = the number chosen correctly


=      r! /                   x                   (n-r)! /

     k! (k-r)!                        [(n-r)-(r-k)]! x (r-k)!


=   5! /                    x            (47-5)! /

   4! (5-4)!                      [(47-5)-(5-4)]! x (5-4)!


= 5 x 4! /               x             42!/

  4! x 1!                           (42 - 1)! x (1!)


= 5                       x            42 x 41! / 



= 5 x   42


= 210


Probability of getting 4/5 numbers correct and Mega number correct:

= total probabilities /

    4/5 coefficient


= 41,416,353 /



= 197,221


This example shows that in 197,221 tickets sold, there will be at least one ticket that will match 4/5 numbers and match the Mega number correctly.


Can you increase your chances of winning by buying multiple tickets?

Person filling in bubbles on a lottery ticket


Yes! By dividing the odds of the game you want to win by the number of tickets you purchase you can calculate your new odds of winning. For example, if you purchase 10,000 tickets and are trying to win the 4/5 match and Mega number match your odds of winning the game that day are one in 19.7 or approximately a 5% chance of winning that prize that day.


For your consideration: the odds of winning only apply to an individual game. Just like a flipping a coin has a 50/50 chance for heads on each toss, you must apply the same logic to this lottery game. Your 5% chance of winning only applies if you purchase 10,000 tickets for one game, not over a period of time for multiple games.


Let’s look again at the lesser prizes to get an idea of how many people play the lottery on a given day.

For Feburary 11, 2023 there was:

1 winner for the Match 5, with an odds of winning at 1 in 1,592,937

18 winners for the 4 Match + Mega, with an odds of winning at 1 in 197,221

373 winners for the Match 4 out of 5, with an odds of winning at 1 in 7,585

631 winners for the Match 3 out of 5 + mega, with an odds of winning at 1 in 4,810

16,019 winners for the Match 3, with an odds of winning at 1 in 185

7,890 winners for the Match 2 + Mega, with an odds of winning at 1 in 361

36,859 winners for the Match 1 + Mega, with an odds of winning at 1 in 74

55,488 winners for the Mega only, with an odds of winning at 1 in 49


We can estimate the number of players that day by multiplying the odds of winning by the total number of winners for each case, then take the average.

Win Type Estimated Tickets Sold

Match 5 = 1,592,937              

Mach4 4 + Mega = 3,549,978

Match 4 = 2,829,205

Match 3 + Mega = 3,035,110

Match 3 = 2,963,515

Match 2 + Mega = 1,459,650

Match 1 + Mega = 2,875,566

Match Mega only = 2,718,912


Average = 2,783,189.125


If we assume a fair game, then we can also assume the average estimated tickets sold is somewhere around 2.7 million plus or minus 0.5 million. The odds of someone winning go up with the number of tickets sold. However, this does not mean your individual odds improve. 


If it is statistically unlikely to win the lottery, why do people play the game?


This is a question best left for sociologists and psychologists and neurobiologists, as the reasons for gambling vary by individual. From a biological standpoint, it is thought that the desire to play the lottery is a heuristic developed due to a past positive experience with the lottery. At some point, the individual received positive feedback from playing the game, whether that be as simple as wish-fulfillment day-dreaming as to what they would be able to achieve with a large amount of money or as extreme as needing the money to pay off debts. The latter would probably benefit by consulting a debt consolidation company and/or seek financial and/or psychological counseling.

I hope, if you do decide to play the lottery, keep top-of-mind that a good portion of the money will most likely go towards the educational system in your state, though it may only provide for about 1.5% of the total educational budget — at least, in California. But after you purchase your ticket and once your dollar is given to the state it will then trickle back to your community in the form of education funding for your public schools, colleges, state hospitals and prisons.


Bottom Line on Playing the Lottery

3-dimensional lottery balls floating, various numbers can be seen in black paint on white ball

Generally, playing the lottery is a light-hearted way for us to dream big and through our day dreams provide a better future for those seeking greater education. But if as much as 1.5 million people in California have a gambling problem, is it possible that more than half of the daily lottery users actually have a gambling problem too? (Why would you buy one ticket if you could double your chances of winning by purchasing 2 tickets?)

By feeding this desire for winning big are we doing good by those 2.7 million ticket purchases? Or do the ends (1.5% of the state education budget) justify the means (enabling potential problem gamblers)? That’s a tough call.

However, now you have more information to use for the next time you consider purchasing a state lottery ticket. In the end, should you like to increase the odds of someone winning, it may be beneficial to play when other people are playing, usually around major consumer holidays like Halloween, Valentine’s day, etc.



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