# 3 ways to calculate the odds of winning the lottery

Everyone has heard comparisons between the odds of winning the lottery and the odds of other unlikely events (such as being struck by lightning). While it is true that the odds of winning the jackpot in a game like Powerball or another 6-number pick lottery game are extremely low, how low are they? And how many times would you have to play to have a better chance of winning? You can find these answers down to the exact probabilities with some simple calculations.

## Steps

### Method 1 of 3: Calculate the Odds of Winning the Powerball Jackpot

#### Step 1. Understand the calculations involved

If you want to determine the odds of winning any lottery, divide the number of winning numbers by the total number of possible numbers. In case the numbers are chosen from a set and the order doesn't matter, use the formula n! R! (N − r)! { Displaystyle { frac {n!} {R! (N-r)!}}}

. In the formula, n represents the total number of possible numbers and r represents the number of numbers chosen. The sign" 2\times=" />

• Luego, esto se resolvería de la siguiente forma: 5×4×3×2×12×1×3×2×1{displaystyle {frac {5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1}}}
• , que es 120÷12{displaystyle 120\div 12}

, or 10. </li>

<li> Therefore, your chances of winning this game are 1 in 10. </li>

#### Step 2. Establish the lottery rules

Most lotteries like Mega Millions, Powerball, and other large lotteries use roughly the same rules: 5 or 6 numbers are chosen from a large group of numbers in no particular order. It is not possible to repeat the numbers. In some games, a final number is chosen from a smaller set of numbers (an example is the "Powerball" number in Powerball games). In Powerball, 5 numbers are chosen out of 69 possible numbers. Then, for the single Powerball number, a number is chosen from a set of 26 possible numbers.

### Other games may have you choose 5 or 6 numbers, or more from a larger or smaller group of numbers. If you want to calculate the odds of winning, you just need to know the number of winning numbers and the total number of possible numbers

#### Step 3. Plug the numbers into the probability equation

The first part of the Powerball odds determines the number of ways in which 5 numbers could be chosen out of 69 unique numbers. Using the Powerball rules, the completed equation for the first 5 numbers would be 69! 5! (69−5)! { Displaystyle { frac {69!} {5! (69-5)!}}}

, que se simplifica a 69!5!×64!{displaystyle {frac {69!}{5!\times 64!}}}

#### Step 4. Calculate your probabilities of choosing correctly

To solve this equation, it is best to do it in its entirety in a search engine or calculator because it is inconvenient to write down the numbers involved between each step. The result tells you that there are 11 238 513 possible combinations of 5 numbers in a set of 69 unique numbers. This means that your chances are 1 in 11 238 513 of picking all five numbers correctly.

### If you want to calculate your probabilities of choosing the final Powerball number correctly, you would complete the same equation using the values for the Powerball (1 number out of 26 possible numbers). Since you only choose one number here, you don't necessarily have to complete the entire equation. The answer will be 26, since there are 26 different ways a number can be chosen from a set of 26 unique numbers

#### Step 5. Multiply to calculate your chances of winning the pot

If you want to calculate the odds that you will guess the first 5 numbers and the Powerball number correctly to win the jackpot, multiply the odds that you will guess the first 5 numbers (1 in 11 238 513) by the odds that you will correctly guess the Powerball number (1 in 26). The equation would be 111, 238, 513 × 126 { displaystyle { frac {1} {11, 238, 513}} times { frac {1} {26}}}

### Método 2 de 3: Determinar las probabilidades para los premios menores

#### Step 1. Calculate your chances of winning the second prize

Going back to the Powerball game, there are 5 numbers and a single Powerball number. In case you guess the 5 remaining numbers correctly but don't get the Powerball number, you will win the second prize. If you calculated your odds of winning the jackpot, you already know that your odds of guessing all 5 numbers correctly are 1 in 11,238,513.

• To win the second prize, you would have to guess the Powerball number incorrectly. In case you have calculated your odds of winning the jackpot, you know that your odds of guessing the Powerball number correctly are 1 in 26. Therefore, your odds of guessing the Powerball number incorrectly are 25 in 26.
• Use the same equation with these values to determine your odds of winning the second prize: 111, 238, 513 × 2526 { displaystyle { frac {1} {11, 238, 513}} times { frac {25} { 26}}}

. Al terminar este cálculo, verás que tus probabilidades de ganar el segundo premio son de 1 en 11 688 053, 52.

#### Step 2. Use an expanded equation to find your odds of winning other prizes

To win other prizes, you correctly guess some of the winning numbers, but not all of them. If you want to determine your probabilities, use an equation in which "k" represents the numbers you choose correctly, "r" represents the total number of numbers that were obtained, and "n" represents the number of unique numbers from which the numbers will be taken. numbers. Without the numbers, the formula looks like this: r! K! × (r − k)! × (n − r)! ((N − r) - (r − k))! × (r − k)! { displaystyle { frac {r!} {k! \ times (rk)!}} times { frac {(nr)!} {((nr) - (rk))! \ times (rk)!}} }

• Por ejemplo, podrías usar los valores del Powerball para determinar tus probabilidades de adivinar correctamente 3 de los 5 números elegidos del conjunto de 69 números únicos. La ecuación se vería así: 5!3!×(5−3)!×(69−5)!((69−5)−(5−3))!×(5−3)!{displaystyle {frac {5!}{3!\times (5-3)!}}\times {frac {(69-5)!}{((69-5)-(5-3))!\times (5-3)!}}}
• El resultado de esta ecuación te indica la cantidad de formas en las que se pueden elegir 3 números correctamente de 5 números. Tus probabilidades serán ese número de la cantidad total de formas en las que se pueden elegir 5 números correctamente.

#### Step 3. Solve the equation to find the probabilities of guessing the numbers correctly

As with the base equation, it is best to solve this equation by typing it all into a calculator or search engine. It would be cumbersome to write down some of the intermediate numbers involved in the calculation and it would be easy to make a mistake.

### In the example above, your odds of guessing 3 of the 5 chosen numbers on the Powerball would be 20 160 out of 11 238 513

#### Step 4. Multiply the result by the value of the Powerball number to determine your odds of winning that prize

This formula gives you the odds of correctly guessing only some of the numbers, but you have not yet taken the Powerball number into consideration. If you want to find your true odds, multiply the result by the odds of correctly or incorrectly guessing the Powerball number (whatever value you want to get).

• For example, if you want to calculate your chances of correctly guessing only 3 of the 5 numbers and not hitting the Powerball number, the equation would be 20, 16011, 238, 513 × 2526 { displaystyle { frac {20, 160} {11, 238, 513}} times { frac {25} {26}}}

, o de 1 en 579, 76.

• Por otro lado, las probabilidades de adivinar correctamente 3 de los 5 números y acertar en el número Powerball serían 20, 16011, 238, 513×126{displaystyle {frac {20, 160}{11, 238, 513}}\times {frac {1}{26}}}
• , o de 1 en 14 494, 11.

#### Step 5. Change the number of numbers you correctly guessed for other prizes

After mastering the formula, just change the value of "k" to find the odds of winning prizes of different levels. In general, your chances of winning will decrease as the value of "k" increases.

### Method 3 of 3: Calculate the Odds of Winning Other Lotteries

#### Step 1. Find the expected return on a lottery ticket

The expected return tells you what you could theoretically expect to get back for buying a single lottery ticket. If you want to calculate the expected return on a single ticket, multiply the odds of a particular payment by the value of that payment. If you do it for every prize you could win, you will get a range of expected returns.

#### Step 3. Determine the increase in odds by playing multiple times

Playing the lottery multiple times increases your overall odds of winning, no matter how small. It is easier to visualize this increase as a decrease in your chances of losing.

• For example, if your overall odds of winning are 1 in 250,000,000, your odds of losing on one spin are 249,999,999 ÷ 250,000,000 { displaystyle 249,999,999 \ div 250,000,000}

. Esto equivale a un número muy cercano a 1 (0, 99999…).

• En caso de que juegues dos veces, ese número se eleva al cuadrado ((249, 999, 999÷250, 000, 000)2{displaystyle (249, 999, 999\div 250, 000, 000)^{2}}
• ), lo que representa un alejamiento ligero de 1 (y, por ende, mejores probabilidades de ganar).

#### Step 4. Find the number of plays it takes to have a good chance of winning

For the most part, those who play the lottery are convinced that if they play often enough, they will have a significantly higher chance of winning. While it is true that playing more increases the odds of winning, it takes a long time for that increase in odds to become significant.

• For example, if you had a 1 in 250,000,000 chance of winning on one spin, it would take about 180 million spins to arrive at a 50-50 chance of winning.
• At this rate, if you bought 10 tickets a day for 49,300 years, you would have a 50% chance of winning.
• Also, should you finally hit the 50-50 odds, anyway, you wouldn't be guaranteed to win if you bought two tickets that day. Your overall odds of winning would still be approximately 50% for each of those tickets.