Last updated on January 2, 2024

**Do you know how many tickets you must buy to get a 50/50 chance of winning a prize? Let’s explore the fascinating world of probability calculations for a random lottery game together.**

Table of Contents

## Inverse Thinking

We commonly solve for the probability of winning the lottery, but sometimes, it’s easier to invert the process to get a different take on the problem.^{1}

For example, instead of calculating the probability of winning, we calculate the probability of losing according to the number of tickets we buy. We subtract the result from one to get the probability of winning a prize.

Let’s use the Canada Lotto 6/49 game as an example.^{2}

## The Probability of Winning Nothing

The chance of winning the jackpot prize in a Lotto 6/49 game is 1 in 13,983,816 possibilities. So, the probability is 0.00000007151.

P_{(winning the jackpot)} = 1/13,983,816

To increase your chances of winning, you buy more tickets. So, if you buy two tickets, the chances that you hit the jackpot will be 2 in 13,983,816 or a probability of 0.00000014302.

The closer you get to the value of 1, the better your chances of winning the grand prize.

Of course, we also use the probability tool to know the likelihood that something will not happen.

For example, we can ask a question like:

*“What is the likelihood that two tickets will not win any prize?”*

We answer the above question by calculating the probability of winning nothing.

According to the official website of the Canada Lotto 6/49 game, the odds of winning any prize are 1 in 6.6.^{3}.

So, the probability that a single ticket is not any prizewinner is 0.84848484845. If you buy more tickets, the probability of losing with *n* number of tickets is this number raised to the *n*th power.

Therefore, your probability of losing for buying two tickets is 0.71978256, or 72%.

*P _{(losing by buying two tickets)} = 0.8485^{2}*

Now that you know how to calculate the probability of winning nothing, let’s find out how many tickets you need to buy to have a 50-50 chance of winning any prize.

## The 50-50 Chance of Winning Any Prize

When you buy more tickets, your probability of losing decreases while your probability of winning increases. A decrease on one side of the equation will increase the other side’s value.

Therefore, by subtracting the probability of winning nothing from 1, whatever remains indicates the probability of winning any prize.

*P _{(winning any prize)} = 1 – P_{(losing)}^{[number of tickets]}*

If you calculate *1 – 0.8485 ^{4}*, you’ll get 0.481668757719937. Subtract that from one, and you’re close to getting a 50% chance of winning any prize. So, buying five tickets for each game gives you a 56% chance of winning any prize.

*P _{(winning any prize)} = 1 – 0.8485^{5} = 56%*

## Plotting the 50-50 Chance of Winning

The image below indicates the 50-50 mark where the two lines intersect.

In perspective, to give you 99.99% certainty that you will win any prize, you must buy 56 tickets.

*1 – 0.8485 ^{56} = 0.999898957655978 = 99.99%*

Note that with this 99.99%, you will likely win a “free play” based on the Canada Lotto’s payout scheme. This is the mathematical expectation because the probability favors the lowest-tier prize.^{4}

## How To Win a Lottery Jackpot

If someone claims you can win small prizes frequently while you wait for the big one to happen. Please, don’t believe.

The expected value (EV) of the lottery is negative.^{5}

Even if the jackpot prize increases, the EV always stays negative. This negative expectation highlights the fact that you can’t profit from a lottery game.^{6}

Mathematically speaking, to win the lottery, you must have long patience.

How long? Let’s calculate.

Each time you play the Canada Lotto 6/49 game with one ticket, your probability of losing is *13,983,815/13,983,816*. That’s a 99.999992848% guarantee of losing. So your probability of losing twice in a row is this number squared.

*P _{(losing the jackpot twice)} = 0.99999992848^{2} = 0.999999856977528*

If you play 9,692,842 times, you will reach that 50/50 chance of winning the jackpot prize. So, assuming you play one ticket once a week, you must live for 186,401 years. Please note that we are not talking about a 99.99% chance.

*0.99999992848 ^{9,692,842 tickets} = 0.499957549546977*

Winning the jackpot prize is difficult to achieve.

If you want to win a lottery jackpot, I suggest some tips you can apply immediately. My article, How to Win the Lottery According to Math, should be able to help you.

## More Advanced Mathematical Studies of the Lottery

If you want to use mathematics to win the lottery, I strongly suggest you avoid statistics and use combinatorial mathematics and probability theory as your main tools. Please read The Winning Lottery Formula Using Math.

However, if math is not your thing, I recommend using a Lotterycodex calculator.

Our discussion reinforces how difficult it is to win a lottery game. Aside from being an expensive hobby over time, there’s no guarantee. That explains why playing the lottery is not an alternative to a full-time job.

Please play the lottery for fun.

## Questions and Answers

**What does ‘inverse thinking’ mean in lottery probability?**

Inverse thinking in lottery probability refers to focusing on the odds of not winning rather than the odds of winning. This approach emphasizes understanding the high probability of losing to develop a more realistic perspective on lottery games.

**What is the mathematical expectation in lottery jackpots?**

The mathematical expectation in lottery jackpots is generally negative. This means that, on average, a player loses money when they buy a lottery ticket. The expectation value considers the prize’s size versus the probability of winning.

**How do combinatorial mathematics and probability theory influence lottery strategies?**

Combinatorial mathematics and probability theory provide a more accurate framework for understanding lottery odds than statistical methods. They focus on the success-to-failure ratio of combinatorial groups, offering a more precise approach to lottery strategy.

## Additional Resources

- Inversion: The Crucial Thinking Skill Nobody Ever Taught You [↩]
- How to Win Lotto 6/49 According To Math [↩]
- The Official Odds and Payout of Canada Lotto 6/49 [↩]
- Probability Theory [↩]
- Expected Value in Lotteries: Exposing Its Significance and Impact [↩]
- The Truth About Winning Small Prizes in the Lottery [↩]

Yeah four five years ago 6/49 was 6-10 mln now 60 and all conditions same. So there is something with furnished lottery 🤔

The jackpot prize has nothing to do with the probability that’s why the condition is the same.

How do you win the lottery not how are you done with the lottery

Hello Antonio, the scope of your inquiry is quite extensive for the article. If you’re genuinely interested in learning the mathematical approach to winning the lottery, I recommend checking out a dedicated article on how to win the lottery at https://lotterycodex.com/how-to-win-the-lottery-mathematically/.