dependent_probability.html

{% extends "multiple_choice_template.html" %}
{% block maincode %}
<script language=Javascript1.2><!--
// Dependent Probability
// Author: Joey Chang
// Date: 1/19/11
//
// Problem layout:
// You have A coins in a bag. B of them are unfair in that they have a C% chance of coming up heads when flipped (the rest
// are fair coins). You randomly choose one coin from the bag and flip it D times. What is the probability of getting D heads?
//
// A - coins in bag
// B - unfair coins
// C - % chance of unfair coins coming up heads
// D - number of times chosen coin is flipped
//


var ExerciseDependentProbability = {

    // constant values
    COINS_MIN: 5,			// random range for total coins in bag
    COINS_MAX: 15,			// .
    UNFAIR_COINS_MIN: 2,		// random range for unfair coins (NOTE: MAX must be <= COINS_MIN)
    UNFAIR_COINS_MAX: 4,		// .
    PERCENT_UNFAIR_HEADS_MIN: 30,	// random range for % chance heads shows on the unfair coin
    PERCENT_UNFAIR_HEADS_MAX: 70,	// .
    PERCENT_UNFAIR_HEADS_INCR: 5,	// increment of selected random value (i.e. 20, 25, 30, 35... are possible values for incr of 5)
    COIN_FLIPS_MIN: 2,			// random range for number of coin flips
    COIN_FLIPS_MAX: 4,			// .
    COLOR_FAIR: "#33CC33",		// color used to tag fair-related terms in hints
    COLOR_UNFAIR: "#CC3333",		// color used to tag unfair-related terms in hints

    numCoins: 0,			// randomly chosen number of coins in bag
    numUnfairCoins: 0,			// randomly chosen number of unfair coins
    percentChanceUnfairHeads: 0,	// randomly chosen % chance of flipping heads with unfair coin
    numFlips: 0,			// randomly chosen number of flips
    answer: 0,				// computed answer (saved for use in hint system)
    
    init: function() {
        this.numCoins = getRandomIntRange(this.COINS_MIN, this.COINS_MAX);
	this.numUnfairCoins = getRandomIntRange(this.UNFAIR_COINS_MIN, this.UNFAIR_COINS_MAX);
	var numIncr = (this.PERCENT_UNFAIR_HEADS_MAX - this.PERCENT_UNFAIR_HEADS_MIN) / this.PERCENT_UNFAIR_HEADS_INCR;
	this.percentChanceUnfairHeads = this.PERCENT_UNFAIR_HEADS_MIN + getRandomInt(numIncr) * this.PERCENT_UNFAIR_HEADS_INCR;
    if(this.percentChanceUnfairHeads == 50){
        this.percentChanceUnfairHeads = 50 + this.PERCENT_UNFAIR_HEADS_INCR;  //Prevent the unfair coin from being fair
    }
	this.numFlips = getRandomIntRange(this.COIN_FLIPS_MIN, this.COIN_FLIPS_MAX);

	// compute answer
	this.computeAnswer();

	// create some convenience variables for forming the hint text
	var unfairCoinFractionString = this.numUnfairCoins + '/' + this.numCoins;
	var fairCoinFractionString = (this.numCoins-this.numUnfairCoins) + '/' + this.numCoins;
	var decimalChanceUnfairHeads = this.percentChanceUnfairHeads / 100;
	var unfairHeadsDecimalFormula = decimalChanceUnfairHeads.toString();
	for (var i = 0; i < this.numFlips - 1; i++)
	{
		unfairHeadsDecimalFormula += '*' + decimalChanceUnfairHeads;
	}
	var unfairHeadsPercentFormula = this.percentChanceUnfairHeads + '%';
	for (var i = 0; i < this.numFlips - 1; i++)
	{
		unfairHeadsPercentFormula += '*' + this.percentChanceUnfairHeads + '%';
	}
	var decimalChanceFairHeads = 0.5;
	var fairHeadsDecimalFormula = decimalChanceFairHeads.toString();
	for (var i = 0; i < this.numFlips - 1; i++)
	{
		fairHeadsDecimalFormula += '*' + decimalChanceFairHeads;
	}
	var fairHeadsPercentFormula = this.percentChanceFairHeads + '%';
	for (var i = 0; i < this.numFlips - 1; i++)
	{
		fairHeadsPercentFormula += '*' + this.percentChanceFairHeads + '%';
	}
	
	// write question text
	write_text('You have ' + this.numCoins + ' coins in a bag. ' + this.numUnfairCoins + ' of them are ' +
		   'unfair in that they have a ' + this.percentChanceUnfairHeads + '% chance of coming up heads when ' +
		   'flipped (the rest are fair coins). You randomly choose one coin from the bag and flip it ' +
		   this.numFlips + ' times. What is the percent probability of getting ' + this.numFlips + ' heads?');
	
	// hint indicating different coin types
	write_step('<br>You can only possibly pick a fair coin or pick an unfair coin. There is no other outcome.');

	// hint questioning coin type selection chances
	write_step('<br>What chance do you have of picking an unfair coin? How about a fair coin?');

	// hint showing chance of unfair coin picked
	write_step('<br>An unfair coin occurs ' +
		   mathFormatWithColor(unfairCoinFractionString, this.COLOR_UNFAIR) +
		   ' of the time.');

	// hint showing chance of fair coin picked
	write_step('<br>A fair coin occurs the rest of the time, or ' +
		   mathFormatWithColor(fairCoinFractionString, this.COLOR_FAIR) +
		   ' of the time.');

	// hint questioning head-flipping chances for unfair coin
	write_step('<br>For that ' +
		   mathFormatWithColor(unfairCoinFractionString, this.COLOR_UNFAIR) +
		   ' of the time that you pick an unfair coin, what is the chance of flipping ' +
		   mathFormatWithColor(this.numFlips, this.COLOR_UNFAIR) +
		   ' heads using that unfair coin?');

	// hint showing chance of flipping all heads for unfair coin
	write_step('<br>The chance is ' +
		   mathFormatWithColor(unfairHeadsPercentFormula, this.COLOR_UNFAIR) +
		   ', or ' +
		   mathFormatWithColor(unfairHeadsDecimalFormula, this.COLOR_UNFAIR) +
		   '.');

	// hint suggesting combining unfair coin selection with all-heads-flip event
	write_step('<br>Now, then, your chance of both picking the unfair coin and also flipping ' +
		   mathFormatWithColor(this.numFlips, this.COLOR_UNFAIR) +
		   ' heads--the chance that both these events occur--is what?');

	// hint revealing computed chance with unfair coin
	write_step('<br>It is ' +
		   mathFormatWithColor(unfairCoinFractionString + '*' + unfairHeadsDecimalFormula, this.COLOR_UNFAIR));
		   
	// hint questioning the fair coin computation	
	write_step('<br>Now, the other possibility, picking the fair coin and flipping ' +
		   mathFormatWithColor(this.numFlips, this.COLOR_FAIR) +
		   ' heads is what?');

	// hint showing the fair coin computation	
	write_step('<br>It is ' +
		   mathFormatWithColor(fairCoinFractionString + '*' + fairHeadsDecimalFormula, this.COLOR_FAIR));

	// hint questioning how to combine previously computed events
	write_step('<br>How do you combine these two mutually exclusive events to find the chance that either occurs?');

	// hint revealing the answer
	write_step('<br>Add them! So your answer is ' +
		   mathFormatWithColor(unfairCoinFractionString + '*' + unfairHeadsDecimalFormula, this.COLOR_UNFAIR) +
		   ' + ' +
		   mathFormatWithColor(fairCoinFractionString + '*' + fairHeadsDecimalFormula, this.COLOR_FAIR) +
		   ', or ' +
		   this.answer +
		   '%.');
    },

    // Computes correct answer and adds wrong choices within a reasonable max range given the selected random inputs
    computeAnswer: function() {
	// the answer is chanceUnfairPicked * (unfairCoinHeads ^ flips) + chanceFairPicked * (.5 ^ flips)
	var chanceUnfairPicked = this.numUnfairCoins / this.numCoins;
	var chanceFairPicked = 1 - chanceUnfairPicked;
	
	// answer in rounded xx.xx% form, i.e. 32.63
	this.answer = Math.round(10000 * (chanceUnfairPicked * Math.pow((this.percentChanceUnfairHeads / 100), this.numFlips) +
		chanceFairPicked * Math.pow(.5,this.numFlips))) / 100;

	setCorrectAnswer(this.answer + '%');
	
	// cap the wrong answers at a maximum value
	var maxAnswerSeed = Math.pow(Math.max(50,this.percentChanceUnfairHeads) / 100, this.numFlips) * 10000;
	
	addWrongChoice(getRandomInt(maxAnswerSeed) / 100 + '%');
	addWrongChoice(getRandomInt(maxAnswerSeed) / 100 + '%');
	addWrongChoice(getRandomInt(maxAnswerSeed) / 100 + '%');
	addWrongChoice(getRandomInt(maxAnswerSeed) / 100 + '%');	
    },

}

// initialize the exercise!
ExerciseDependentProbability.init();

//-->
</script>
{% endblock maincode %}

{% block graphdisplay %}
<td valign=top><iframe name="present" frameborder=0 src="/graphpage.html?w=260&h=260" height="290" width="290"></iframe></td>
{% endblock graphdisplay %}

{% block maincellwidth %}70%{% endblock maincellwidth %}