A little math probablility problem.
I was originally planning tomorrow to post the new Perspective Magazine for you to ogle at (distribution on Friday!) but I shall delay that and you shall be rewarded for waiting with a PDF you can download of the magazine. Ooh.
Instead tonight I had some free time (because tomorrow I shall be skiving school spending time productively at home) So I decided to look at an NRICH problem. Yes, that’s right. I was so bored I decided to do math. Edit: Apparently I didn’t notice the star rating system so it seems as though I picked an easy one.
Click to see problem: Succession in Randomia
Let’s consider a probability tree:
… OK. The first thing we notice is that it looks prettier. Let’s see the series for B now: 1/4 + 1/16 + 1/64 + … . This is identical to T. We can do the same for L and say we have 2/8 + 2/32 + 2/128 … which is exactly the same as B and T. Plugging it into a/1-r (r being 1/4) we get 1/3 for all three. Hence we can say that in fact it is a completely unbiased way of choosing a successor.
My answer to that question is therefore: “Yeah they have an equal chance”
Well, I wasn’t going to stop here. Why don’t I simulate it? Here is the coding that simulates the situation (the cheapest brute force technique to tackle the problem):
If you don’t know programming, go take a read through the code anyway and see if you can get a grasp of what’s going on :)
<?php
// The number of times each king as won.
$bingo = 0;
$toto = 0;
$lotto = 0;
// How many times we are going to do the crowning ceremony
$ceremonies = 10000;
// Loop through these instructions to carry out the crowning ceremony.
for ($i = 1; $i < $ceremonies+1; $i++)
{
$win = FALSE; # Nobody has become king yet.
$oddeven = 1; # Because the first throw is odd. Assume 1 = odd, 0 = even.
$firsttoss = TRUE; # We are tossing for the first time.
// This variable is used later to determine which variable $x or $y is
// reassigned a value and which keeps the previous toss value.
$a = 0;
// Keep on tossing the coins until somebody wins the kingdom.
while ($win == FALSE)
{
// Here we only flip one coin, this will alternate between $x and $y.
// The one ($x or $y) that isn't assigned a new value will retain the
// previous toss value so we can find out whether or not we have got
// two HH or TT in a row.
// Assume H = 1, T = 2
if ($a == 0) {
$x = rand(1,2);
$a = $a + 1;
} elseif ($a > 0) {
$y = rand(1,2);
$a = 0;
}
if ($firsttoss == TRUE) {
// Nobody can win on the first toss.
$firsttoss = FALSE;
} else {
// If it is HH and ODD...
if ($x == 1 && $y == 1 && $oddeven == 1) {
// Lotto will become King.
$lotto = $lotto + 1;
// Somebody has won the Kingdom.
$win = TRUE;
}
// If it is TT and ODD...
if ($x == 2 && $y == 2 && $oddeven == 1) {
// Lotto will become King.
$lotto = $lotto + 1;
// Somebody has won the Kingdom.
$win = TRUE;
}
// If it is HH and EVEN...
if ($x == 1 && $y == 1 && $oddeven == 0) {
// Bingo will become King.
$bingo = $bingo + 1;
// Somebody has won the Kingdom.
$win = TRUE;
}
// If it is TT and EVEN...
if ($x == 2 && $y == 2 && $oddeven == 0) {
// Toto will become King.
$toto = $toto + 1;
// Somebody has won the Kingdom.
$win = TRUE;
}
}
// ... If nobody has won anything ...
if ($win != TRUE) {
// We flip again, so Odd->Even or Even->Odd.
if ($oddeven == 0) {
$oddeven = 1;
} else {
$oddeven = 0;
}
}
}
}
// Tell the computer to print out what results we have.
echo 'Bingo: '. $bingo .'
';
echo 'Toto: '. $toto .'
';
echo 'Lotto: '. $lotto .'
';
?>
I ran it and here is an example result I get. Out of 10,000 ceremonies, this is the number of times each king has won:
Bingo: 3322
Toto: 3340
Lotto: 3338
Pretty close, eh?
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