Coin Toss Test

In a room containing 1024 people, what are the odds that at least one person in the room could correctly call 10 coin tosses in a row? Here are some random trails for that scenario. After each flip all the winners go on to the next round.

For this purpose, a number is selected at random between 0 and 1023. That number is converted to binary, each binary digit representing one of the ten coin flips, zero being tails and one being heads. Thus, all the coin flips for any of the 1024 possible numbers are predetermined, though our 1024 individuals in the room will be flipping their coins without knowing the outcome. The process of flipping the coins proceeds left to right using the generated binary number. If the left-most digit is zero, then the first flip will have been determined to be tails; if one, the first flip will be heads. After each flip all the winners go on to the next round. The program takes the number of individuals left from the last previous round and performs a "flip" that many times, producing a one or zero at random for each. If the binary digit being tested in our 10-digit binary number is one then all the individuals who had been assigned a one will win in that round. There may or may not be individuals left standing after the tenth coin flip.
 

Trial for 209
209 = 0011010001

Flip 1—tails: 522 of 1024
Flip 2—tails: 269 of 522
Flip 3—heads: 138 of 269
Flip 4—heads: 69 of 138
Flip 5—tails: 42 of 69
Flip 6—heads: 15 of 42
Flip 7—tails: 6 of 15
Flip 8—tails: 3 of 6
Flip 9—tails: 1 of 3
Flip 10—heads: 1 of 1

  

<?php
ini_set( 'error_reporting', E_ERROR ) ;
mt_srand( intval(substr(microtime(),4,3)) + 1 ) ;
$sd = mt_rand( 50, 200 ) ;
$im = $_REQUEST[ 'im' ] ;
?>
<div style="border: 1px solid #C60000 ; background-color: #DEDEFF ; margin: 20px ; " >
<img src="https://franklinseiberling.com/img/transparentHorizontal.png" alt="550" width="550" height="1" >
<div style="font: 14px Verdana ; margin: 20px ; " >
<b>Coin Toss Test</b><br>
<p style="font: 11px Verdana ; color: #606060 ; " >
<br>In a room containing 1024 people, what are the odds that at least one person in the room could correctly call 10 coin tosses in a row? Here are some random trails for that scenario. After each flip all the winners go on to the next round.<br><br>
For this purpose, a number is selected at random between 0 and 1023. That number is converted to binary, each binary digit representing one of the ten coin flips, zero being tails and one being heads. Thus, all the coin flips for any of the 1024 possible numbers are predetermined, though our 1024 individuals in the room will be flipping their coins without knowing the outcome. The process of flipping the coins proceeds left to right using the generated binary number. If the left-most digit is zero, then the first flip will have been determined to be tails; if one, the first flip will be heads. After each flip all the winners go on to the next round. The program takes the number of individuals left from the last previous round and performs a "flip" that many times, producing a one or zero at random for each. If the binary digit being tested in our 10-digit binary number is one then all the individuals who had been assigned a one will win in that round. There may or may not be individuals left standing after the tenth coin flip.<br>&nbsp;</p>

<?php
if ( ( $im == NULL ) | ( $im > 1023 ) | ( $im < 0 ) | ( !is_numeric( $im ) ) ) { $rn = mt_rand( 0, 1023 ) ; } else { $rn = $im ; }
print ( 'Trial for '.$rn.'<br>' ) ;
$bits = decbin( strval( $rn ) ) ;
$rb = intval( $bits ) ;
while( strlen( $bits ) < 10 ) { $bits = '0'.$bits ; }
print( $rn.' = '.$bits.'<br><br>' ) ;
$ix = 1 ;
$H_T = array() ;
while( $ix <= 10 ) {
$H_T[ $ix ] = $rb - ( intval( $rb / 10 ) ) * 10 ;
$rb = intval( $rb / 10 ) ;
$ix ++ ;
}
$fc = 10 ;
$tf = 1024 ;
while( $fc > 0 ) {
$nr = 0 ;
$ix = 1 ;
while( $ix <= $tf ) {
$flip = mt_rand( 0, 1 ) ;
if ( $flip == $H_T[ $fc ] ) { $nr ++ ; }
$ix ++ ;
}
print( 'Flip '.( 11 - $fc ).'&mdash;'.( ( $H_T[ $fc ] == "1" )? "heads" : "tails" ).': '.$nr.' of '.($ix - 1).'<br>' ) ;
$fc = $fc - 1 ;
$tf = $nr ;
}
?>
</div>

<div align="center" style="margin-bottom: 10px ; " >
<input type="button" value="&nbsp;New Trial&nbsp;" onClick="javascript: window.location.assign( 'http://franklinseiberling.com/?pg=coinToss' ) ; ">
&nbsp;&nbsp;<input type="button" value="&nbsp;New Trial&mdash;Same Flips&nbsp;" onClick="javascript: window.location.assign( 'http://franklinseiberling.com/?pg=coinToss&amp;im=<?php print( $rn ) ; ?>' ) ; " >
</div>

</div>

<div align="center" >
<img src="https://franklinseiberling.com/img/transparentHorizontal.png" alt="550" width="550" height="1" ><br>
<a href="javascript: ToggleDisplay( 'sourceT' ) ; " class="w" >View/Hide source code</a>
<br>
</div>

<div id="sourceT" style="display: none ; " >
<?php
$filePath = $jw.'/FranklinSeiberling/rightPanel/coinToss.php' ;
viewSource( $filePath ) ;
?>
</div>