Do You Really Have a Chance to Win the Lottery?

Mathematics can help you calculate the probability of winning and decide what’s more advantageous: buying 10 lottery tickets for one draw or purchasing one ticket for 10 different draws.

By Denis Peshekhonov

With a master's degree in engineering, Denis develops software and occasionally builds mathematical models for games.

In the American TV series "Numb3rs," the main character is a mathematician who assists the FBI in solving crimes. In one episode, he states that the chance of being killed on the way to buy a lottery ticket is higher than the chance of winning the lottery. At the end of this article, I will share calculations related to this claim. But first, let's explore the math behind popular gambling games and how it might slightly improve your odds.

Rule 1: Evaluate Your Risks

It’s no secret that casinos and gambling establishments design their games to ensure they always profit. They do this by offering payouts that are less favorable than the odds of winning. Simply put, if you bet $1 and the payout is $1,000, your chance of winning is less than 1 in 1,000.

There are no exceptions unless someone intentionally wants to give you money. Keep this simple rule in mind to maintain a realistic perspective.

Game theory assesses any strategy by multiplying the probability of winning by the payout size. For example, getting $1,000 guaranteed is mathematically equivalent to a 50% chance of winning $2,000. This principle helps you compare games roughly: is it better to have a $1 million prize with a 1 in 100,000 chance or $50 with a 1 in 4 chance? Intuitively, the first seems more appealing, but mathematically, the second is more advantageous.

Strictly mathematically, winning at a casino is impossible because the product of win probability and payout is always less than your bet.

However, people gamble not only for money but also for the excitement and thrill of the game and winning.

Additionally, money’s value isn’t linear: receiving $1 immediately is like having a 1 in 1,000,000 chance to get $1 million. Losing $1 usually doesn’t affect life, but winning $1 million is a life-changing event.

Rule 2: Play Transparently

Unfortunately, we can’t peek behind the scenes of lotteries, but understanding their official procedures is useful.

For example, classic slot machines, known as "one-armed bandits," can be deceptive. The visible wheel shows symbols with various values, leading players to believe each symbol has an equal chance of appearing. In reality, whether mechanically or digitally, the actual reels hidden behind have valuable symbols appearing rarely and cheap ones frequently.

The odds of hitting triple 7s are much lower than hitting three cherries — sometimes by a factor of ten or more.

"Transparent" lotteries are fairer. In the US, tickets contain either a sequence of numbers or let players choose their own. In Russia, a lotto format is common, where players must match one or all lines of numbers to win.

While theoretically, organizers could print losing tickets or rig the draw, reputable companies don’t risk scandals because they already profit from the game.

If you plan to gamble, it’s wise to understand how the game works and ensure no external manipulation affects the results.

Rule 3: Know Your Odds

The jackpot probability in most lotteries is calculated by a standard formula. Calculating the chance to win smaller prizes, like matching a single line in lotto, is more complex and beyond this article’s scope. However, most lotteries offer secondary prizes, increasing your chances of winning something.

Let’s focus on the jackpot for simplicity.

Suppose you buy a lottery ticket with a random set of numbers. During the draw, the same number of balls is pulled, and if your numbers match those drawn (in any order), you win. The probability is:

Winning Probability = 1 ÷ Number of possible ball combinations.

The number of combinations without order is known in math as combinations. If you’re not familiar, online calculators can help. These require:

• n — total number of possible numbers (balls) in the lottery;
• k — number of balls drawn (and numbers on your ticket).

For example, a lottery drawing 5 balls from 50 numbers (1 to 50) has odds of:

1 ÷ 2,118,760 = 0.00005%.

Consider the popular US PowerBall lottery, where jackpots have exceeded $1 billion. The rules require matching 5 numbers from 1 to 69 plus one additional number from 1 to 26.

The chance to match the first 5 numbers is 1 in 11,238,513, and the additional ball is 1 in 26. Multiplying these gives:

(1 ÷ 11,238,513) × (1 ÷ 26) = 1 ÷ 292,201,338 = 0.0000003%.

In other words, if 300 million tickets are sold, one person wins the jackpot. This explains why jackpots often roll over — organizers don’t print enough tickets to guarantee a winner.

Rule 4: Start Playing at the Right Time

PowerBall tickets cost $2. To calculate when buying a ticket becomes worthwhile, multiply the ticket price by the odds denominator:

2 × 292,201,338 = $584,402,676

Remember to factor in taxes—winners usually receive about 70% of the jackpot. So, the jackpot must be at least $850 million to mathematically justify buying a ticket.

How does this align with the earlier statement that the expected value is usually negative? Because if no one wins, the jackpot rolls over and grows, accumulating until it reaches a point where buying a ticket is mathematically beneficial.

Ideally, you should skip all draws until the jackpot reaches that threshold, then buy tickets for that specific draw.

However, predicting when a jackpot will roll over is impossible. Still, starting to buy tickets once the jackpot exceeds this amount makes sense mathematically.

Which is better: buying many tickets for one draw or one ticket for multiple draws? Let’s analyze.

In probability theory, independent events don’t influence each other. For example, rolling two dice results are independent; the outcome of one doesn’t affect the other. Conversely, drawing two cards from a deck without replacement involves dependent events because the first card affects the second.

The gambler’s fallacy is a common misunderstanding where people believe past independent events influence future outcomes. For instance, after many coin tosses landing heads, one might expect tails to be more likely next, but probabilities remain constant.

In lotteries, different draws are independent events since the balls are reshuffled each time. Your chance of winning a particular draw doesn’t depend on previous plays. This is hard to accept intuitively because players often think, "I’ve played so many times; I’m due for a win!" But probability doesn’t work that way.

Buying multiple tickets for the same draw increases your chances proportionally because tickets are mutually exclusive combinations. Buying 10 tickets with different numbers increases your odds 10-fold compared to buying one ticket.

So, if you have money for 10 tickets, it’s better to buy them all for one draw rather than one ticket for 10 draws.

Still, the chance of winning at least once in multiple draws is slightly higher than winning in a single draw but slightly lower than buying multiple tickets for one draw.

If you buy a ticket monthly for fun, the process itself might matter more than odds. Mathematically, saving for a year and buying 12 tickets at once is better, though losing that way might feel more disappointing.

Rule 5: Know When to Stop

Even a 1 in 100 chance is very low. Checking such odds monthly means 100 tries over 8 years. Imagine how much lower 1 in 1,000,000 or 1 in 100,000,000 odds are. Always gamble only with money you can afford to lose entirely.

Finally, to address the initial claim about the risk of being killed on the way to buy a lottery ticket versus winning, here are some US-based statistics.

In 2016, approximately 17,000 murders occurred in the US. Assuming a 50-year risk period for an adult, that totals 850,000 murders. With a population of 325.7 million, the chance of being a victim is:

850,000 ÷ 325,700,000 = 1 ÷ 383 = 0.3%.

Assuming you buy a ticket once a month, and you leave home 6 days a week (26 days a month), the risk per ticket-buying trip is:

(1 ÷ 383) ÷ 26 = 1 ÷ 9,958 = 0.01%.

Even this rough estimate shows that the risk of being killed on the way to buy a lottery ticket is significantly higher—about 30,000 times more likely—than winning the jackpot.

Of course, actual numbers vary based on factors like lifestyle and demographics, but the principle remains.

Living without hope for good outcomes or expecting constant misfortune, even knowing the math, isn’t the best choice.

*Note: Meta Platforms Inc. and its social networks Facebook and Instagram are restricted in Russia.

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