Calculating Powerball Odds with PowerShell

Overview

There was recently a Powerball jackpot that soared to over a billion dollars, capturing the nation’s attention and dominating headlines. As news stations broadcast the staggering jackpot amount, excitement spread like wildfire, and people from all walks of life rushed to purchase their tickets, hoping to strike it rich. Amidst the frenzy, I found myself at a local Wawa, urged by friends to buy tickets. Admittedly, I didn’t even know how to play Powerball at the time. Standing there, holding those lottery tickets, I couldn’t shake the feeling that I was essentially dropping money into a bottomless pit, hoping against hope for a miracle. It’s a peculiar feeling, the blend of anticipation and uncertainty that accompanies the purchase of a Powerball ticket. Needless to say the anticipation was stifled when I learned about the statistical chances of actually winning.

How Does the Powerball work?

The Powerball game is structured around sixty-nine white balls, each bearing a numerical value ranging from one to sixty-nine. Out of these, five white balls are drawn, along with a single red “Power Ball” chosen from twenty-six balls numbered one to twenty-six. To claim the coveted jackpot, your ticket must match all six drawn numbers. Beyond the jackpot, there are other ways to win. Matching the red Powerball or having two or more correct white ball numbers can still land you a prize. For a detailed breakdown of how the game operates and its various payout options, check out this comprehensive article by USAToday.

Calculating the Odds of Winning the Powerball

Odds of Hitting Five Balls

The method to calculate the odds of winning heavily depend on factorials. I also posted about string permutation and touched on factorials, as this is a fairly new concept for me. Let’s describe the variables in the function. To calculate the odds of getting all five white balls correct, we would use the function

1. N! (Factorial of 69)
   • Represents all possible combinations for the white balls.
   • It’s the total number of ways to pick 5 numbers out of 69 without considering the order.
2. (N – S)! (Factorial of (69 – 5))
   • Represents the number of ways to choose 5 white balls out of 69.
3. S! (Factorial of 5)
   • Represents the number of ways to arrange the 5 chosen white balls.

When you pass in the values, it should look like this:

Where N = 69 and S = 5

This returns 11,238,513. So the odds of you getting the five white balls are 1 in 11,238,513. This does not include the red ball “PowerBall” between one and twenty-six. Continue to divide the odds by the number of tickets you bought to get an accurate probability, because each ticket has an independent chance of winning.

Odds of Hitting the PowerBall Jackpot

Adding the Powerball, you multiply our current formula by the number of red balls, R, so:

PowerShell Script to Calculate the Odds of Winning the PowerBall

PowerShell Function to calculate Factorials

We declare our necessary variables and write the function calcFactorial. Started identifying ways to use recursion and thought this was a straightforward task.

$whiteBallNumbers = 69
$whiteBallAmount = 5
$redBallNumbers = 26

# Function to calculate factorials
function calcFactorial {
    [CmdletBinding()]
    param (
        [Parameter(Mandatory=$true, HelpMessage="A non-negative integer")]
        [int] $n,

        [Parameter(HelpMessage="The current factorial being calculated. (Default 1)")]
        $sum = [System.Numerics.BigInteger]::One
    )

    if($n -eq 0){ return $sum }

    $sum *= $n
    $n--
    calcFactorial -n $n -sum $sum
}

Call the Function and Format the Output

This article discusses in-depth string formatting so we can output our large number in human-readable format, comma separated by the thousands.

# Calculate the odds
$whiteBallTwoCombo = (calcFactorial -n $whiteBallNumbers) / ((calcFactorial -n ($whiteBallNumbers - 2)) * (calcFactorial -n 2))
$whiteBallFiveCombo = (calcFactorial -n $whiteBallNumbers) / ((calcFactorial -n ($whiteBallNumbers - $whiteBallAmount)) * (calcFactorial -n $whiteBallAmount))
$redBallCombo = (calcFactorial -n $redBallNumbers) / (calcFactorial -n ($redBallNumbers - 1))

Write-Output ("2 white ball odds: 1 in {0:n0}" -f $whiteBallTwoCombo)
Write-Output ("5 white ball odds: 1 in {0:n0}" -f $whiteBallFiveCombo)
Write-Output ("Red PowerBall odds: 1 in {0:n0}" -f $redBallCombo)

$powerBallCombo = $whiteBallFiveCombo * $redBallCombo

Write-Output ("Hittng the Powerball: 1 in {0:n0}" -f $powerBallCombo)

Just a quick note, in the PowerShell code I used the function F(N,S) for our Red Powerball calculation and stored it in the variable $redBallCombo, but because there is only one in the set, it will return 26, so this can be set as a constant or you can use $redBallNumbers defined at the top of our script. I also tested our formula on the probability of selecting two correct white balls. You can continue to test or refactor the code. Although this is simple in terms of mathematics, it is really cool to conceptualize a problem, use a mathematical formula, and apply it with code.

Conclusion

In delving into the mechanics of Powerball, we uncover a game of staggering odds and tantalizing possibilities. With sixty-nine white balls, five of which are drawn, and an additional twenty-six red “Power Balls”, the number combinations quickly skyrocket into the millions. Through the lens of mathematics, we’ve calculated that your chances of matching all five white balls stand at 1 in 11,238,513, and for the red Powerball, it’s 1 in 26. When you multiply these two odds together, you’re looking at a staggering 1 in 292,201,338 chance of hitting the Powerball jackpot. These numbers serve as a sobering reminder of the sheer randomness of the game, but they also underscore the thrill that keeps millions lining up for a chance at that life-changing jackpot. Whether you’re a seasoned player or a curious newcomer, understanding the odds adds a fascinating layer to the Powerball experience.