# Khan/khan-exercises

Reviewers: eater

Reviewed By: eater

CC: emily

Differential Revision: http://phabricator.khanacademy.org/D531
1 parent 65c3bc4 commit 260074bff5eb4241c9428c0d8bded3d19eaddeb5 mwittels committed Aug 8, 2012
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+ randFromArray([ + ["a 1", 1], + ["a 2", 1], + ["a 3", 1], + ["a 4", 1], + ["a 5", 1], + ["a 6", 1], + ["a 7", 1], + ["an 8", 1], + ["a 9", 1], + ["a 10", 1], + ["at least a 2", 9], + ["at least a 5", 6], + ["at least a 7", 4], + ["more than a 2", 8], + ["more than a 6", 4], + ["more than an 8", 2], + ["less than a 4", 3], + ["less than a 7", 6], + ["less than an 8", 7], + ["an even number", 5], + ["an even number", 5], + ["an odd number", 5], + ["an odd number", 5] + ]) + 10 - MAKE_COUNT + fraction(MAKE_COUNT,10,true,false) + fraction(LOSE_COUNT,10,true,false) + randRange(5,10) + randRange(5,10) + MAKE_COUNT*MAKE - LOSE_COUNT*LOSE + + [fraction(PROFIT,10,true,false), + (PROFIT/10).toFixed(2)] + +
+ +

+ A game at the carnival offers these odds: you get to roll a + ten-sided die, and if you roll RESULT_DESC, + you make MAKE dollars. Unfortunately, + if you roll anything else, you lose + LOSE dollars. +

+ +

+ How much money do you expect to make (or lose) + playing this game? +

+ +
+ $+ ANS + + + + + The expected value of an event (like playing this game) + is average of the values of each outcome. Since some + outcomes are more likely than others (sometimes), we + weight the value of each outcome according to its + probability to get an accurate idea of what value + to expect. + + + There are two events that can happen in this game: either + you roll RESULT_DESC, or you don't. So, the + expected value will look like this: + + E = + (money gained when you roll RESULT_DESC) + \cdot + (probability of rolling RESULT_DESC) + + + (money gained when you don't roll RESULT_DESC) + \cdot + (probability of not rolling RESULT_DESC). + + + The money you gain when you win + is$MAKE. + The probability of winning is the probability + that you roll RESULT_DESC. +

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+ This probability is the number of winning outcomes + divided by the total number of + outcomes, MAKE_FR. +

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+ The money you gain when you lose is + $-LOSE + (since you actually lose money). The probability that + you lose is the probability that you don't roll + RESULT_DESC. + + + This probability must be + 1 - MAKE_FR = LOSE_FR. + + + So, if we take the average of the amount of money you make + on each outcome, weighted by how probable each outcome is, + we get the expected amount of money you will make: + (MAKE\cdotMAKE_FR) + + (-LOSE\cdotLOSE_FR) = +$ANS_F = $ANS. + + + + + + + randFromArray([4,6,10,12]) + + (function(){ + if(SIDES < 7) { + return _.map(_.range(SIDES), function(i){ + return "\\dfrac{"+(i+1)+"}{"+SIDES+"}"; }) + .join("+"); + } + + first = _.map(_.range(3), function(i){ + return "\\dfrac{"+(i+1)+"}{"+SIDES+"}"; }) + .join("+"); + last = _.map(_.range(3), function(i){ + return "\\dfrac{"+(SIDES-2+i)+"}{"+SIDES+"}"; }).join("+"); + return [first,"\\cdots",last].join("+"); + })() + + + _.reduce(_.range(SIDES), function(n,i){ return n+i+1; }, 0) + + + + If you roll a SIDES-sided die, what is the expected + value you will roll? + + + + ANS_N/SIDES + + + + + The expected value of an event (like rolling a die) + is average of the values of each outcome. To get an + accurate idea of what value of expect, we + weight the value of each outcome according to its + probability. + + + In this case, there are SIDES outcomes: + the first outcome is rolling a 1, the second outcome is + rolling a 2, and so on. The value of each of these outcomes + is just the number you roll. + + + So, the value of the first outcome is 1, and its + probability is \dfrac{1}{SIDES}. + + + The value of the second outcome is 2, the value of + the third outcome is 3, and so on. There are + SIDES outcomes altogether, and each of them + occurs with probability + \dfrac{1}{SIDES}. + + + So, if we average the values of each of these outcomes, + we get the expected value we will roll, which is + SUM = + mixedFractionFromImproper(ANS_N,SIDES,true,true). + + + + + + random() < 0.4 + randRange(2,4) + randRange(1,5)*100 + BUY ? + COST*ODDS + randRange(1,3)*100 : + COST*ODDS - randRange(1,3)*100 + + fraction(1,ODDS,true,true) + BUY ? + "Yes, the expected value is positive." : + "No, the expected value is negative." + + + + + You decide you're only going to buy a lottery ticket if the + expected amount of money you will get is positive. Tickets + cost$COST, and you get + $PRIZE if you win. The odds of + winning are 1 in ODDS, + meaning that you will win with probability + ODD_F. + + + + Should you buy a ticket for this lottery? + + + + ANS + + + + • Yes, the expected value is positive. • + • No, the expected value is negative. • + + + + + The expected value of an event (like buying a lottery + ticket) is the average of the values of each outcome. + In this case, the outcome where you win is much less likely + than the outcome that you lose. So, to get an accurate idea + of how much money you expect to win or lose, we have to + take an average weighted by the probability of each outcome. + + + As an equation, this means the expected amount of money + you will win is + E = (money gained when you win) + \cdot (probability of winning) + + (money gained when you lose) + \cdot (probability of losing) + . + + + Let's figure out each of these terms one at a time. The + money you gain when you win is your winnings minus the + cost of the ticket,$PRIZE - + $COST (you may find the math easier + if you don't simplify this). + + + From the question, we know the probability of winning is + ODD_F. + + + The money you gain when you lose is actually negative, + and is just the cost of the ticket, + -$COST. +

+

+ Finally, the probability of losing is (1 - + ODD_F) (you may find the math + easier if you don't simplify this). +

+

+ Putting it all together, the expected value is + E = ($PRIZE -$COST) + (ODD_F) + (-$COST) + (1 - ODD_F) = +$ \dfrac{PRIZE} + {ODDS} - $\cancel{\dfrac{COST} + {ODDS}} -$COST + + $\cancel{\dfrac{COST}{ODDS}} = +$fraction(PRIZE,ODDS,true,true) - + $COST. + + +$fraction(PRIZE,ODDS,true,true) - + \$COST is + PRIZE/ODDS - COST > 0 ? "positive" : "negative". +

+ So, we expect to PRIZE/ODDS - COST > 0 ? "make" : "lose" money by buying a lottery ticket, because + the expected value is PRIZE/ODDS - COST > 0 ? + "positive" : "negative". +

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