Calculate the possibility of minimum correct answers in single-choice questions with 4 choices (by just guessing!)

answer-possibility.php

<?php 

bcscale(70);

function factorial($number){

	$factorial = 1;

	while($number > 0){
		$factorial = bcmul($factorial, $number);
		$number--;
	}

	return $factorial;

}

function compound($select, $total) {
	return bcdiv(factorial($total), bcmul(factorial($select), factorial($total - $select)));
}

function possibility($total, $minimum) {
	
	$sum = 0;

	for($i = $minimum; $i <= $total; $i++){
		$sum = bcadd(
			bcmul(
				bcmul(
				compound($i, $total),
				bcdiv(1, bcpow(4, $i))
				),
				bcpow(bcdiv(3, 4), $total - $i)
			),
			$sum
		);
	}

	return $sum;
}

$total = $argv[1];
$minimum = $argv[2];

$possibility = possibility($total, $minimum);
echo "Possibility: " . possibility($total, $minimum) . PHP_EOL;
echo "Point: " . ((($minimum * 3) - ($total - $minimum)) * 100) / ($total * 3) . PHP_EOL;

Examples:

If we have 4 questions, the possibility of guessing at least 2 correct answers is:

$ php answer-possibility.php 4 2
Possibility: 0.2617187500000000000000000000000000000000000000000000000000000000000000

100 questions, at least 50 correct answers:

php answer-possibility.php 100 50
Possibility: 0.0000000663850248347665153516542686470914346302257991431899859403687471

100 questions, at least 20 correct answers:

php answer-possibility.php 100 20
Possibility: 0.9004695898946859448449903562740041725413774125137943152056172746750182

Please study hard instead of guessing!!