FORMULA 1 LOTTERY SYSTEM: lottery ticket

Though there's no perfect method for choosing winning lottery numbers, you can experiment with a few different ways to make the process fun. Here are four different ways to fill in your lucky lottery ticket!

Method One: Frequency Picks

1 Look up the frequency chart for previous lottery draws. Most lotteries will offer charts showing how often each number has been drawn during a given timeframe. (For example, you can see a Powerball frequency chart via the Iowa State Lottery here).

Choose your numbers based on the frequency chart. After you've looked over the odds, you have two options:
- Select numbers that are drawn frequently. If you notice that a few numbers stand out for being drawn significantly more often than the others, consider including them in your pick. Be aware, though, that several other people will be trying this tactic; if you win with frequently-picked numbers, you might be at a greater risk of having to share the prize with other winners.
- Select numbers that are drawn less frequently. Including numbers that aren't picked very often might seem like a counterintuitive strategy, but consider this: if everyone else is busy picking frequently drawn numbers and you win with your long-shot picks, you might not have to share the prize with as many other winners.
3
Be aware that each number still has an equal chance of being drawn. Looking at frequency charts might show you which numbers tend to be drawn, but keep in mind that when the actual lottery drawing comes around, each number still has an absolutely equal chance of being picked.

Method Two: The Delta System

For this example we will use numbers 1 through 50. Make sure your upper limit is the same as the lottery you're playing.

1
Choose a very low number. If you feel good about the number 1, choose 1. Winning numbers sometimes have two consecutive numbers, but sometimes they don't. There is no way to predict a winning number.
2
Pick two numbers between 1 and 8. For example, 3 and 5.
3
Pick a number very close to 8. For example, 9.
4
Pick two numbers between 8 and 15. For example, 11 and 13.
5
Write down your delta numbers. Our numbers are 1-3-5-9-11-13.
6
Mix up the delta numbers. For example, 5-3-11-9-1-13.
7
Write down the first delta number. This is our first lottery number.
8
Add the first lottery number and the second delta number together. This is the second lottery number. Repeat this for the rest of the delta numbers.
- For this example, your lottery numbers will be 5-8-19-28-29-42.
- This method only works for drawings with 6 winning numbers.
- Make sure you add you delta numbers up to make sure the total is lower that the highest number in the drawing. (example, 1+3+5+9+11+13=42).

Method Three: Lucky Numbers

1
Choose numbers that are significant to you. If you believe in lucky numbers, they're probably digits that center around important dates or events in your life. For example, you might use:
- Birthdays: Yours, your children's, your spouse's, and so on.
- Anniversaries: This could be a wedding anniversary date, or the date of another significant event.
- Ages: Using your age or the ages of your loved ones is also a common practice.
- Addresses: The address of your childhood or current home is another tactic you can try.
- Phone numbers: Try breaking down your phone number into a sequence of single- or double-digit lotto numbers.
2
Choose numbers you consider lucky. Some folks have a lucky number they use for everything, that isn't connected to anything like a birthday. If this is the case for you, add your lucky number to the mix!
- If you're playing a lottery like Powerball, consider making your lucky number the powerball pick.
- Be aware that most people consider numbers like 7 and 11 lucky, and know that they're extremely common lottery picks. If you win with these numbers on your ticket, you might end up sharing the prize with a lot of people.

Method Four: Random Numbers

1
Find a random number generator. Random.org has one specifically designed for lottery picks.
- Enter your variables.
- Write down your numbers on your lottery ticket.
2
Or, allow the lottery to choose randomly for you. You can request random picks when you purchase your lottery ticket. If you're buying several tickets at one time, this is probably the way to go.
Tips
- Remember, the lottery is completely random and there's no perfect way to pick numbers. If there were, it wouldn't be fun!
Soure: wikihow.com

Lottery mathematics is used here to mean the calculation of the probabilities in a lottery game. The lottery game used in the examples below is one in which one selects 6 numbers from 49, and hopes that as many of those 6 as possible match the 6 that are randomly selected from the same pool of 49 numbers in the "draw".

Calculation explained in choosing 6 from 49

In a typical 6/49 game, six numbers are drawn from a range of 49 and if the six numbers on a ticket match the numbers drawn, the ticket holder is a jackpot winner—this is true no matter in which order the numbers appear. The probability of this happening is 1 in 13,983,816.

This small chance of winning can be demonstrated as follows:

Starting with a bag of 49 differently-numbered lottery balls, there are 49 different but equally likely ways of choosing the number of the first ball selected from the bag, and so there is a 1 in 49 chance of predicting the number correctly. When the draw comes to the second number, there are now only 48 balls left in the bag (because the balls already drawn are not returned to the bag) so there is now a 1 in 48 chance of predicting this number.

Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but of course we could have gotten to this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49, again in the correct order, is 1 in 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as ${49!\over (49-6)!}$ . This works out to a very large number, 10,068,347,520, which is much bigger than the 14 million stated above.

The last step is to understand that the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any set of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 720 orders in which they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as 49! / (6! × (49 - 6)!), or more generally as

${n\choose k}={n!\over k!(n-k)!}$ .

This function is called the combination function; in Microsoft Excel, this function is implemented as COMBIN(n, k). For example, COMBIN(49, 6) (the calculation shown above), would return 13,983,816. For the rest of this article, we will use the notation ${n\choose k}$ . "Combination" means the group of numbers selected, irrespective of the order in which they are drawn.

An alternative method of calculating the odds is to never make the erroneous assumption that balls must be selected in a certain order. The odds of the first ball corresponding to one of the six chosen is 6/49; the odds of the second ball corresponding to one of the remaining five chosen is 5/48; and so on. This yields a final formula of

${n\choose k}={49\choose 6}={49\over 6} * {48\over 5} * {47\over 4} * {46\over 3} * {45\over 2} * {44\over 1}$

The range of possible combinations for a given lottery can be referred to as the "number space". "Coverage" is the percentage of a lottery's number space that is in play for a given drawing.

Odds of getting other possibilities in choosing 6 from 49

One must divide the number of combinations producing the given result by the total number of possible combinations (for example, ${49\choose 6} = 13,983,816$ , as explained in the section above). The numerator equates to the number of ways one can select the winning numbers multiplied by the number of ways one can select the losing numbers.

For a score of n (for example, if 3 of your numbers match the 6 balls drawn, then n = 3), there are ${6\choose n}$ ways of selecting n winning numbers from the 6 winning numbers. This means that there are 6 - n losing numbers, which are chosen from the 43 losing numbers in ${43\choose 6-n}$ ways. The total number of combinations giving that result is, as stated above, the first number multiplied by the second. The expression is therefore ${6\choose n}{43\choose 6-n}\over {49\choose 6}$ .

This can be written in a general form for all lotteries as: ${K\choose B}{N-K\choose K-B}\over {N\choose K}$ , where $N$ is the number of balls in lottery, $K$ is the number of balls in a single ticket, and $B$ is the number of matching balls for a winning ticket.

The generalisation of this formula is called the hypergeometric distribution (the HYPGEOMDIST() function in most popular spreadsheets).

This gives the following results:

Score	Calculation	Exact Probability	Approximate Decimal Probability	Approximate 1/Probability
0	${6\choose 0}{43\choose 6}\over {49\choose 6}$	435,461/998,844	0.436	2.2938
1	${6\choose 1}{43\choose 5}\over {49\choose 6}$	68,757/166,474	0.413	2.4212
2	${6\choose 2}{43\choose 4}\over {49\choose 6}$	44,075/332,948	0.132	7.5541
3	${6\choose 3}{43\choose 3}\over {49\choose 6}$	8,815/499,422	0.0177	56.66
4	${6\choose 4}{43\choose 2}\over {49\choose 6}$	645/665,896	0.000969	1,032.4
5	${6\choose 5}{43\choose 1}\over {49\choose 6}$	43/2,330,636	0.0000184	54,200.8
6	${6\choose 6}{43\choose 0}\over {49\choose 6}$	1/13,983,816	0.0000000715	13,983,816

Powerballs And Bonus Balls

Many lotteries have a powerball (or "bonus ball"). If the powerball is drawn from a pool of numbers different from the main lottery, then simply multiply the odds by the number of powerballs. For example, in the 6 from 49 lottery, if there were 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 × 10, or 566.6 (the probability would be divided by 10, to give an exact value of 8815/4994220). Another example of such a game is Mega Millions, albeit with different jackpot odds.

Where more than 1 powerball is drawn from a separate pool of balls to the main lottery (for example, in the Euromillions game), the odds of the different possible powerball matching scores should be calculated using the method shown in the "other scores" section above (in other words, treat the powerballs like a mini-lottery in their own right), and then multiplied by the odds of achieving the required main-lottery score.

If the powerball is drawn from the same pool of numbers as the main lottery, then, for a given target score, one must calculate the number of winning combinations, including the powerball. For games based on the Canadian lottery (such as the United Kingdom's lottery), after the 6 main balls are drawn, an extra ball is drawn from the same pool of balls, and this becomes the powerball (or "bonus ball"), and there is an extra prize for matching 5 balls and the bonus ball. As described in the "other scores" section above, the number of ways one can obtain a score of 5 from a single ticket is ${6\choose 5}{43\choose 1}$ or 258. Since the number of remaining balls is 43, and the ticket has 1 unmatched number remaining, 1/43 of these 258 combinations will match the next ball drawn (the powerball). So, there are 258/43 = 6 ways of achieving it. Therefore, the odds of getting a score of 5 and the powerball are ${6}\over {49\choose 6}$ = 1 in 2,330,636.

Of the 258 combinations that match 5 of the main 6 balls, in 42/43 of them the remaining number will not match the powerball, giving odds of ${258 \cdot {{42}\over {43}}}\over {49\choose 6}$ = 3/166,474 (approximately 55,491.33) for obtaining a score of 5 without matching the powerball.

Using the same principle, to calculate the odds of getting a score of 2 and the powerball, calculate the number of ways to get a score of 2 as ${6\choose 2}{43\choose 4}$ = 1,851,150 then multiply this by the probability of one of the remaining four numbers matching the bonus ball, which is 4/43. Since 1,851,150 × (4/43) = 172,200, the probability of obtaining the score of 2 and the bonus ball is ${172,200}\over {49\choose 6}$ = 1025/83237. This gives approximate decimal odds of 81.2.

The general formula for $B$ matching balls in a $N$ choose $K$ lottery with one bonus ball from the $N$ pool of balls is: ${K-B\over N-K}{K\choose B}{N-K\choose K-B}\over {N\choose K}$

The general formula for $B$ matching balls in a $N$ choose $K$ lottery with zero bonus ball from the $N$ pool of balls is: ${N-K-K+B\over N-K}{K\choose B}{N-K\choose K-B}\over {N\choose K}$

The general formula for $B$ matching balls in a $N$ choose $K$ lottery with one bonus ball from a separate pool of $P$ balls is: ${1\over P}{K\choose B}{N-K\choose K-B}\over {N\choose K}$

The general formula for $B$ matching balls in a $N$ choose $K$ lottery with no bonus ball from a separate pool of $P$ balls is:

${P-1\over P}{K\choose B}{N-K\choose K-B}\over {N\choose K}$

Minimum number of tickets for a match

It is a hard, in most cases open, mathematical problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers. In the 5-from-90 lotto, the minimum number that can guarantee a ticket with at least 2 matches is 100.

FORMULA 1 LOTTERY SYSTEM

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Formula 1 Lotto System

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1/21/13

How to Choose Lottery Numbers