In a simple lottery where 6 numbers are drawn from a range of 49 and if the 6 numbers on a ticket match the numbers drawn and the order of these numbers is disregarded, the ticket holder is a jackpot winner. The probability of this happening is approximately 1 in 14 million (13,983,816 to be exact). Each time a ball is drawn, it is not returned, therefore making the range of numbers available for the next ball drawn is lessen by 1 each time. Therefore in this example, there is clearly a 1 in 49 chance of predicting the number of the first ball drawn, a 1 in 48 chance of predicting the second number, and so on. This means that each of the 49 ways of choosing the first number has 48 ways of choosing the second, thus the chance of correctly predicting the first two numbers is 48x49 or 1 in 2352. Using this premise, the chance of correctly predicting all six numbers is 49 x 48 x 47 x 45 x 44 or 1 in 10,068,347,520.
The formula for this equation can be written like this:49! / (49-6)!
This is not however the final formula for calculating the probability of choosing the correct numbers. The last step is realizing that the order of the numbers is not significant. Any set of 6 numbers has 6! or 720 ways of being drawn, and therefore 10,068,347,520 must be divided by 720 and so the probability of choosing the correct six numbers is 13,983,816.
And so the final formula is : (n/k) = n! / k!(n-k)!
A number of possible combinations for a given lottery can be taken as a class and referred to as the "number space." "Coverage" is the percentage of the number space that is actually in play for a given drawing. Coverage can often be less than 90%.
Many lotteries have a bonus ball, this changes the probability calculations. If the bonus ball is drawn from a different group of numbers from the main lottery, then one simply multiplies the odds by the number of bonus balls. If the bonus ball is drawn from the same pool of numbers as the main lottery, one must calculate the number of winning combinations including the bonus ball. |