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<!DOCTYPE html>
<html data-require="math word-problems math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Expected value</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="game" data-weight="2">
<div class="vars">
<var id="RESULT_DESC, MAKE_COUNT">randFromArray([
["a &lt;code&gt;1&lt;/code&gt;", 1],
["a &lt;code&gt;2&lt;/code&gt;", 1],
["a &lt;code&gt;3&lt;/code&gt;", 1],
["a &lt;code&gt;4&lt;/code&gt;", 1],
["a &lt;code&gt;5&lt;/code&gt;", 1],
["a &lt;code&gt;6&lt;/code&gt;", 1],
["a &lt;code&gt;7&lt;/code&gt;", 1],
["an &lt;code&gt;8&lt;/code&gt;", 1],
["a &lt;code&gt;9&lt;/code&gt;", 1],
["a &lt;code&gt;10&lt;/code&gt;", 1],
["at least a &lt;code&gt;2&lt;/code&gt;", 9],
["at least a &lt;code&gt;5&lt;/code&gt;", 6],
["at least a &lt;code&gt;7&lt;/code&gt;", 4],
["more than a &lt;code&gt;2&lt;/code&gt;", 8],
["more than a &lt;code&gt;6&lt;/code&gt;", 4],
["more than an &lt;code&gt;8&lt;/code&gt;", 2],
["less than a &lt;code&gt;4&lt;/code&gt;", 3],
["less than a &lt;code&gt;7&lt;/code&gt;", 6],
["less than an &lt;code&gt;8&lt;/code&gt;", 7],
["an even number", 5],
["an even number", 5],
["an odd number", 5],
["an odd number", 5]
])</var>
<var id="LOSE_COUNT">10 - MAKE_COUNT</var>
<var id="MAKE_FR">fraction(MAKE_COUNT, 10, true, false)</var>
<var id="LOSE_FR">fraction(LOSE_COUNT, 10, true, false)</var>
<var id="MAKE">randRange(5, 10)</var>
<var id="LOSE">randRange(5, 10)</var>
<var id="PROFIT">MAKE_COUNT * MAKE - LOSE_COUNT * LOSE</var>
<var id="ANS_F">fraction(PROFIT, 10, true, false)</var>
<var id="ANS">PROFIT / 10</var>
</div>
<p>
For a game at the carnival you get to roll a ten-sided die, numbered from <code>1</code> to <code>10</code>.
If you roll <var>RESULT_DESC</var>, you win <code>$<var>MAKE</var></code>.
Unfortunately, if you roll anything else, you lose <code>$<var>LOSE</var></code>.
</p>
<p class="question">
How much money do you expect to make (or lose) per game?
</p>
<div class="solution" data-type="multiple">
<code>$</code>
<span class="sol" data-forms="integer, decimal"><var>ANS</var></span>
</div>
<div class="hints">
<p>
The expected value is a weighted average of the possible values with
the weights determined by the probability of observing that value.
</p>
<p>
There are two events that can happen in this game: either
you roll <var>RESULT_DESC</var>, or you don't. So, the
expected value will look like this:
<br><br>
<code>E = </code>
(money gained when you roll <var>RESULT_DESC</var>)
<code>\cdot</code>
(probability of rolling <var>RESULT_DESC</var>)
<code>+</code>
(money gained when you don't roll <var>RESULT_DESC</var>)
<code>\cdot</code>
(probability of not rolling <var>RESULT_DESC</var>).
</p>
<p>
The money you gain when you win is <code>$<var>MAKE</var></code>.
The probability of winning is the probability
that you roll <var>RESULT_DESC</var>.
</p>
<p>
This probability is the number of winning outcomes
divided by the total number of
outcomes, <code><var>MAKE_FR</var></code>.
</p>
<p>
The money you gain when you lose is <code>-$<var>LOSE</var></code>
(since you actually lose money). The probability that
you lose is the probability that you don't roll
<var>RESULT_DESC</var>.
</p>
<p>
This probability must be
<code>1 - <var>MAKE_FR</var> = <var>LOSE_FR</var></code>.
</p>
<p>
So, if we take the average of the amount of money you make
on each outcome, weighted by how probable each outcome is,
we get the expected amount of money you will make:
<code>(<var>MAKE</var>\cdot<var>MAKE_FR</var>) +
(-<var>LOSE</var>\cdot<var>LOSE_FR</var>) =
<var>ANS_F</var> = <span data-if="ANS < 0">-$<var>localeToFixed(-ANS, 2)</var></span>
<span data-else="">$<var>localeToFixed(ANS, 2)</var></span>.
</code>
</p>
</div>
</div>
<div id="die" data-weight="1">
<div class="vars">
<var id="SIDES">randFromArray([4,6,10,12])</var>
<var id="SUM">
(function(){
if(SIDES &lt; 7) {
return _.map(_.range(SIDES), function(i){
return "\\dfrac{"+(i+1)+"}{"+SIDES+"}"; })
.join("+");
}
first = _.map(_.range(3), function(i){
return "\\dfrac{"+(i+1)+"}{"+SIDES+"}"; })
.join("+");
last = _.map(_.range(3), function(i){
return "\\dfrac{"+(SIDES-2+i)+"}{"+SIDES+"}"; }).join("+");
return [first,"\\cdots",last].join("+");
})()
</var>
<var id="ANS_N">
_.reduce(_.range(SIDES), function(n,i){ return n+i+1; }, 0)
</var>
</div>
<p class="question">
If you roll a <var>SIDES</var>-sided die, what is the expected
value you will roll?
</p>
<div class="solution" data-forms="mixed, improper, decimal">
<var>ANS_N/SIDES</var>
</div>
<div class="hints">
<p>
The expected value is a weighted average of the possible values with
the weights determined by the probability of observing that value.
</p>
<p>
In this case, there are <var>SIDES</var> outcomes:
the first outcome is rolling a 1, the second outcome is
rolling a 2, and so on. The value of each of these outcomes
is just the number you roll.
</p>
<p>
So, the value of the first outcome is 1, and its
probability is <code>\dfrac{1}{<var>SIDES</var>}</code>.
</p>
<p>
The value of the second outcome is 2, the value of
the third outcome is 3, and so on. There are
<var>SIDES</var> outcomes altogether, and each of them
occurs with probability
<code>\dfrac{1}{<var>SIDES</var>}</code>.
</p>
<p>
So, if we average the values of each of these outcomes,
we get the expected value we will roll, which is
<code><var>SUM</var> =
<var>mixedFractionFromImproper(ANS_N,SIDES,true,true)</var></code>.
</p>
</div>
</div>
<div id="lottery" data-weight="2">
<div class="vars">
<var id="BUY">random() &lt; 0.4</var>
<var id="COST">randRange(2,5)</var>
<var id="ODDS">randRange(1,5)*100</var>
<var id="PRIZE">BUY ?
COST*ODDS + randRange(1,3)*100 :
COST*ODDS - randRange(1,3)*100
</var>
<var id="ODD_F">fraction(1,ODDS,true,true)</var>
<var id="ANS">BUY ?
"Yes, the expected value is positive." :
"No, the expected value is negative."
</var>
</div>
<p>
You decide you're only going to buy a lottery ticket if your
expected winnings are larger than the ticket price. Tickets
cost <code>$<var>COST</var></code>, and you get
<code>$<var>PRIZE</var></code> if you win. The odds of
winning are <code>1</code> in <code><var>ODDS</var></code>,
meaning that you will win with probability
<code><var>ODD_F</var></code>.
</p>
<p class="question">Should you buy a ticket for this lottery?</p>
<div class="solution"><var>ANS</var></div>
<ul class="choices" data-category="true">
<li>Yes, the expected value is positive.</li>
<li>No, the expected value is negative.</li>
</ul>
<div class="hints">
<p>
The expected value is a weighted average of the possible values with
the weights determined by the probability of observing that value.
</p>
<p>
This means the expected value, considering both the price
of the ticket and the possible winnings is
<code>E = </code> (money gained when you win)
<code>\cdot</code> (probability of winning) <code>+</code>
(money gained when you lose)
<code>\cdot</code> (probability of losing).
</p>
<p>
Let's figure out each of these terms one at a time. The
money you gain when you win is
<code>$<var>PRIZE</var></code> and from the question, we
know the probability of winning is
<code><var>ODD_F</var></code>.
</p>
<p>
When you lose, you gain no money, or
<code>$0</code>, and the probability of losing is <code>1
- <var>ODD_F</var></code>.
</p>
<p>
Putting it all together, the expected value is
<code>E = ($<var>PRIZE</var>)
(<var>ODD_F</var>) + ($0) (1 - <var>ODD_F</var>) =
$ \dfrac{<var>PRIZE</var>}{<var>ODDS</var>} =
$<var>fraction(PRIZE,ODDS,true,true)</var></code>.
</p>
<p data-if="PRIZE/ODDS - COST > 0">
<code>$<var>fraction(PRIZE,ODDS,true,true)</var> -
$<var>COST</var></code> is positive.
<br><br>
So, we expect to make money by buying a lottery ticket, because
the expected value is positive.
</p><p data-else="">
<code>$<var>fraction(PRIZE,ODDS,true,true)</var> -
$<var>COST</var></code> is negative.
<br><br>
So, we expect to lose money by buying a lottery ticket, because
the expected value is negative.
</p>
</div>
</div>
</div>
</div>
</body>
</html>
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