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add probability with permutations and combinations exercise

Reviewers: eater, emily

Reviewed By: eater

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+<!DOCTYPE html>
+<html data-require="math math-format word-problems subhints">
+ <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
+ <title>Probability with permutations and combinations</title>
+ <script src="../khan-exercise.js"></script>
+ <div class="exercise">
+ <div class="problems">
+ <div id="students">
+ <div class="vars" data-ensure="GROUP &lt; (ARE_B ? BOYS : GIRLS)">
+ <var id="STUDENTS">randRange(7,10)</var>
+ <var id="GROUP">randRange(3,6)</var>
+ <var id="BOYS">randRange(2,STUDENTS-2)</var>
+ <var id="GIRLS">STUDENTS-BOYS</var>
+ <var id="ARE_B">random() &lt; 0.5</var>
+ <var id="NUM_B">ARE_B ? BOYS : GIRLS</var>
+ <var id="ANSWER">(factorial(NUM_B)*factorial(STUDENTS-GROUP))/(factorial(STUDENTS)*factorial(NUM_B-GROUP))</var>
+ <var id="GCD">getGCD(factorial(NUM_B)*factorial(STUDENTS-GROUP),factorial(STUDENTS)*factorial(NUM_B-GROUP))</var>
+ <var id="PRETTY_NUM">factorial(NUM_B)*factorial(STUDENTS-GROUP)/GCD</var>
+ <var id="PRETTY_DEM">factorial(STUDENTS)*factorial(NUM_B-GROUP)/GCD</var>
+ </div>
+ <p>
+ There are <code><var>STUDENTS</var></code> students in a class: <code><var>BOYS</var></code> boys and
+ <code><var>GIRLS</var></code> girls.
+ </p>
+ <p class="question">
+ If the teacher picks a group of <code><var>GROUP</var></code> at random, what is the probability that
+ everyone in the group is a <var>ARE_B ? "boy" : "girl"</var>?
+ </p>
+ <div class="solution" data-forms="proper"><var>ANSWER</var></div>
+ <div class="hints">
+ <p>
+ One way to solve this problem is to figure out how many different groups
+ there are of only <var>ARE_B ? "boys" : "girls"</var>, then divide this
+ by the total number of groups you could choose. Since every groups is chosen
+ with equal probability, this will be the probability that a group of all
+ <var>ARE_B ? "boys" : "girls"</var> is chosen.
+ </p>
+ <p>
+ We know two ways to count the number of groups we can choose: we use permutations if order matters,
+ and combinations if it doesn't. Does the order the students are picked matter in this case?
+ </p>
+ <div>
+ <p>
+ It doesn't matter if we pick <var>ARE_B ? "John" : "Julia"</var> then
+ <var>ARE_B ? "Ben" : "Beatrice"</var> or <var>ARE_B ? "Ben" : "Beatrice"</var> then
+ <var>ARE_B ? "John" : "Julia"</var>,
+ so order must not matter. So, the number of ways
+ to pick a group of <code><var>GROUP</var></code> students out of <code><var>STUDENTS</var></code> is
+ <code> \dfrac{<var>STUDENTS</var>!}{(<var>STUDENTS</var>-<var>GROUP</var>)!<var>GROUP</var>!} =
+ \binom{<var>STUDENTS</var>}{<var>GROUP</var>}</code>.
+ [<a href="#" class="show-subhint" data-subhint="binomial">Show me why</a>]
+ </p>
+ <div class="subhint" id="binomial">
+ <p>
+ Remember that the
+ <code><var>STUDENTS</var>! \;</code> and <code>(<var>STUDENTS</var>-<var>GROUP</var>)! \;</code> terms come from when we
+ fill up the group, making <code><var>STUDENTS</var></code>
+ choices for the first slot, then <code><var>STUDENTS-1</var></code> choices for the
+ second, and so on. In this way, we end up making
+ <code><var>, function(l){ return (STUDENTS-l);}).join("\\cdot")</var>
+ = \dfrac{<var>STUDENTS</var>!}{(<var>STUDENTS</var>-<var>GROUP</var>)!} \;</code>.
+ The <code><var>GROUP</var>! \;</code> term comes from the number of times we've counted
+ a group as different because we chose the students in a different order.
+ There are <code><var>GROUP</var>! \;</code>
+ ways to order a group of <code><var>GROUP</var></code>, so for every group, we've overcounted exactly
+ that many times.
+ </p>
+ </div>
+ </div>
+ <p>
+ We can use the same logic to count the number of groups that only have <var>ARE_B ? "boys" : "girls"</var>.
+ </p>
+ <p>
+ Specifically, the number of ways to pick a group of <code><var>GROUP</var></code> students out of
+ <code><var>NUM_B</var></code> is
+ <code> \dfrac{<var>NUM_B</var>!}{(<var>NUM_B</var>-<var>GROUP</var>)!<var>GROUP</var>!} =
+ \binom{<var>NUM_B</var>}{<var>GROUP</var>}</code>.
+ </p>
+ <p>
+ So, the probability that the teacher picks a group of all <var>ARE_B ? "boys" : "girls"</var> is the number of
+ groups with only <var>ARE_B ? "boys" : "girls"</var> divided by the number of total groups the teacher could pick.
+ </p>
+ <p>
+ This is <code> \displaystyle \frac{\frac{<var>NUM_B</var>!}{(<var>NUM_B</var>-<var>GROUP</var>)!\cancel{<var>GROUP</var>!}}}
+ {\frac{<var>STUDENTS</var>!}{(<var>STUDENTS</var>-<var>GROUP</var>)!\cancel{<var>GROUP</var>!}}} =
+ \frac{\frac{<var>NUM_B</var>!}{<var>NUM_B-GROUP</var>!}}{\frac{<var>STUDENTS</var>!}{<var>STUDENTS-GROUP</var>!}}</code>
+ </p>
+ <p>
+ We can re-arrange the terms to make simplification easier
+ <code>\left(\dfrac{<var>NUM_B</var>!}{<var>NUM_B-GROUP</var>!}\right)
+ \left(\dfrac{<var>STUDENTS-GROUP</var>!}{<var>STUDENTS</var>!}\right) =
+ \left(\dfrac{<var>NUM_B</var>!}{<var>STUDENTS</var>!}\right)
+ \left(\dfrac{<var>STUDENTS-GROUP</var>!}{<var>NUM_B-GROUP</var>!}\right)</code>
+ <p>
+ Simplifying, we get <code>
+ \left(\dfrac{\cancel{<var>NUM_B</var>!}}{<var>, function(l){ return (STUDENTS-l); }).join("\\cdot")
+ </var> \cdot \cancel{<var>NUM_B</var>!}}\right)
+ \left(\dfrac{<var>, function(l){ return (STUDENTS-GROUP-l); }).join("\\cdot")
+ </var> \cdot \cancel{<var>(NUM_B-GROUP)</var>!}}{\cancel{<var>NUM_B-GROUP</var>!}}\right) =
+ \left(\dfrac{1}{<var>factorial(STUDENTS)/factorial(NUM_B)</var>}\right)
+ \left(<var>factorial(STUDENTS-GROUP)/factorial(NUM_B-GROUP)</var>\right) =
+ \dfrac{<var>PRETTY_NUM</var>}{<var>PRETTY_DEM</var>}</code>
+ </p>
+ </div>
+ </div>
+ <div id="coins">
+ <div class="vars">
+ <var id="COINS">randRange(4,7)</var>
+ <var id="NUM">randRange(2,COINS-2)</var>
+ <var id="IS_H">random &lt; 0.5</var>
+ <var id="NAME">IS_H ? "heads" : "tails"</var>
+ <var id="NUM_RIGHT">factorial(COINS)/(factorial(NUM)*factorial(COINS-NUM))</var>
+ <var id="NUM_ALL">Math.pow(2,COINS)</var>
+ <var id="GCD">getGCD(NUM_RIGHT,NUM_ALL)</var>
+ <var id="PRETTY_NUMER">NUM_RIGHT/GCD</var>
+ <var id="PRETTY_DENOM">NUM_ALL/GCD</var>
+ </div>
+ <p class="question">
+ If you flip a fair coin <code><var>COINS</var></code> times, what is the probability that you will get exactly
+ <code><var>NUM</var></code> <var>NAME</var>?
+ </p>
+ <p class="solution" data-forms="proper"><var>PRETTY_NUMER/PRETTY_DENOM</var></p>
+ <div class="hints">
+ <p>
+ One way to solve this problem is to figure out how many ways you can get exactly <code><var>NUM</var></code> <var>NAME</var>, then
+ divide this by the total number of outcomes you could have gotten. Since every outcome has equal probability, this will be the
+ probability that you will get exactly
+ <code><var>NUM</var></code> <var>NAME</var>.
+ </p>
+ <p>
+ How many outcomes are there where you get exactly <code><var>NUM</var></code> <var>NAME</var>? Try thinking of each outcome as
+ a <code><var>COINS</var></code>-letter word, where the first letter is "H" if the first coin toss was heads and "T"
+ if it was tails, and so on.
+ </p>
+ <p>
+ So, the number of outcomes with exactly <code><var>NUM</var></code> <var>NAME</var>
+ is the same as the number of these words which have
+ <code><var>NUM</var></code> <var>IS_H ? "H's" : "T's"</var> and <code><var>COINS-NUM</var></code> <var>IS_H ? "T's" : "H's"</var>.
+ </p>
+ <div>
+ <p>
+ How many of these are there? If we treat all the letters as unique,
+ we'll find that there are <code><var>COINS</var>!</code> different arrangements, overcounting <code><var>NUM</var>!</code>
+ times for every time we only switch the <var>IS_H ? "H's" : "T's"</var> around, and <code><var>COINS-NUM</var>!</code>
+ times for every time we only switch the <var>IS_H ? "T's" : "H's"</var> around.
+ [<a href="#" class="show-subhint" data-subhint="letters">Show me why</a>]
+ </p>
+ <div class="subhint" id="letters">
+ Let's say we toss a coin 5 times, and get tails three times. How many different re-arrangements are there of the
+ letters "HHTTT"? Well, we have five choices for the first slot, four for the second slot, and so on, resulting in
+ <code>5\cdot4\cdot3\cdot2\cdot1 = 5! = 120 \;</code> different re-arrangements. But, this treats all the letters
+ as unique, when </br>
+ H<span class="hint_blue">T</span>H<span class="hint_orange">T</span><span class="hint_green">T</span>
+ is the same as </br>
+ H<span class="hint_orange">T</span>H<span class="hint_blue">T</span><span class="hint_green">T</span>,
+ and </br>
+ H<span class="hint_orange">T</span>H<span class="hint_green">T</span><span class="hint_blue">T</span>,
+ </br>
+ and so on. So really, we need to replace all these different re-arrangements
+ where we only move the tails around with one re-arrangement, HTHTT. There are <code>3!</code> of these multi-
+ colored arrangements for every normal one, so that means dividing
+ our first guess of <code>5!</code> by <code>3!</code>. By the exact same logic, we need to divide by <code>2!</code> to
+ avoid overcounting every permutation where we just move the heads around. So, the number of re-arrangements is
+ <code>\dfrac{5!}{3!2!} = \binom{5}{3}</code>.
+ </div>
+ </div>
+ <div>
+ <p>
+ So, there are <code>\dfrac{<var>COINS</var>!}{<var>NUM</var>!<var>COINS-NUM</var>!} = <var>NUM_RIGHT</var></code> different
+ outcomes where you get exactly <code><var>NUM</var></code> <var>NAME</var>.
+ [<a href="#" class="show-subhint" data-subhint="total">How many total outcomes are there?</a>]
+ </p>
+ <div class="subhint" id="total">
+ Well, if you only flip one coin, there are two outcomes, if you flip two there are four outcomes, if you flip three there
+ are eight. Each time you flip another coin, you double the number of possible outcomes.
+ </div>
+ </div>
+ <p>
+ Altogether, there are <code>2^{<var>COINS</var>} = <var>NUM_ALL</var></code> total possible outcomes.
+ </p>
+ <p data-if="NUM_RIGHT != PRETTY_NUMER">
+ So, the probability that you will get exactly <var>NUM</var> <var>NAME</var> is
+ <code>\dfrac{<var>NUM_RIGHT</var>}{<var>NUM_ALL</var>} = \dfrac{<var>PRETTY_NUMER</var>}{<var>PRETTY_DENOM</var>}</code>.
+ </p>
+ <p data-else>
+ So, the probability that you will get exactly <var>NUM</var> <var>NAME</var> is
+ <code>\dfrac{<var>NUM_RIGHT</var>}{<var>NUM_ALL</var>}</code>.
+ </p>
+ </div>
+ </div>
+ </div>
+ </div>

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