Lottery algorithm - PHP - math seems good, but is the function valid?

Question

I want to make a custom lottery extraction to motivate users to participate in an online experiment. The rules are:

• 10% to get 10$
• 1% chance to get 50$
• 0.1% chance to get 500$

The lottery is a PHP function that gets called once and returns the prize (0, 10, 50 or 500). I created the function below, and after 70 000 trials the statistics are:

• 9.11% for 10$
• .91% for 50$
• .01% for 500$

Should I be worried about the algorithm? Is there a better way to create a good distribution of chances than mt_rand ?

function lottery() {
  // winnings before extraction
  $win=0;

  // choose a winning number between 1 and 10
  $target=mt_rand(1,10);

  // make three independent extractions, each with 1/10 probability
  if (mt_rand(1,10) == $target) {
    // if first extraction is the winning number -> prize=10
    // probability: 1/10
    $win=10;

    if (mt_rand(1,10) == $target) {
        // if second extraction is ALSO the winning number -> prize=50
        // probability: 1/10 * 1/10
        $win=50;

        if (mt_rand(1,10) == $target) {
            // if third extraction is ALSO the winning number -> prize=500
            // probability: 1/10 * 1/10 * 1/10
            $win=500;
        }
    }
  }
  // return the prize
  return $win;
}

Thank you for helping a newbie!

Answer 1

That's because the true chances of getting each in your code are :

$10 - 0.1*0.9 = 9%

$50 - 0.1*0.1*0.9 = 0.9%

$500 - 0.1*0.1*0.1 = 0.1%

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This is not because of mt_rand(). This is just a statistics problem. Try running more iterations, and you'll see the numbers converge to the numbers above.

As you can see with the calculations above, you don't get $10 by getting it right the first time, you get $10 by getting it right the first time (10%), AND getting it wrong the second time (90%).

Following that logic, you can extend the same math for $50 (getting it right the first two times, 10% and 10%, THEN getting it wrong on the third time, 90%), and for $500 (you get the drill).

With your specific code, the computations for the true probability is as above.

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See code in the accepted answer for the correct code with accurate probabilities.

Answer 2

You have four outcomes with given probabilities:

• 0.1% probability to win $500.
• 1% probability to win $500.
• 10% probability to win $500.

The fourth outcome - no win - is 100% minus the sum of the other three outcomes, namely 88.9%.

Mark Gabriel has explained why your initial code was off: By promoting 10% of the people who have won $10 to $50 winners, you take them away from the pool of $10 winner, which will be only 9% of all people.

Pham Trung has contributed a solution that takes the winners of higher amounts from the pool of non-winners so far and adjusts the probabilities. That's a workable solution, but the easiest solution is in my opintion to call the random number generator just once.

This solution also reflects the ticket analogy best: You place 10,000 tickets in a box. The 1,000 tickets from 1 to 1000 win $10. The 100 tickets from 1001 to 1100 win $50. The ten tickets from 1101 to 1110 win $500. All 8890 tickets from 1111 on don't win anything:

function lottery() {
    var pick = Math.floor(10000 * Math.random());
               // random number in the range [0, 10000).

    if (pick < 1000) return 10;
    if (pick < 1100) return 50;
    if (pick < 1110) return 500;

    return 0;
}

In this code, the tickets have only their number written on them. You pick one. Then you check whether the ticket number is eligible for $10. If not, you check the same ticket whether it may bet a $50 win. There is really just one random action involved.

(I've used a zero-based random-number function in Javascript instead of PHP's mt_rand, but I think the solution is clear.)