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Expected value is precisely what you might think it means intuitively : the return you can expect for some kind of action, like how many questions you might get right if you guess on a multiple choice test .
Watch this video recording for a quick explanation of the expected value formula :
Expected Value Formula Watch this television on YouTube
Can’t see the video? Can ’ deoxythymidine monophosphate see the television ? Click here For model, if you take a 20 wonder multiplechoice test with A, B, C, D as the answers, and you guess all “ A ”, then you can expect to get 25 % right ( 5 out of 20 ). The math behind this kind of expected value is :
The probability ( P ) of getting a motion right if you guess : .25
The phone number of questions on the test ( nitrogen ) * : 20
P x n = .25 x 20 = 5
*You might see this as X alternatively .
This character of expected value is called an expected value for a binomial random variable. It ’ s a binomial experiment because there are merely two potential outcomes : you get the answer right, or you get the answer wrong .
The basic expected value formula is the probability of an event multiplied by the amount of times the event happens:
(P(x) * n) .
The formula changes slightly according to what kinds of events are happening. For most simple events, you ’ ll use either the Expected Value formula of a Binomial Random Variable or the Expected Value formula for multiple Events .
The formula for the Expected Value for a binomial random variable is :
P(x) * X.
ten is the number of trials and P ( x ) is the probability of success. For case, if you toss a coin ten times, the probability of getting a heads in each trial is 1/2 so the expected respect ( the number of heads you can expect to get in 10 coin tosses ) is :
P ( x ) * X = .5 * 10 = 5
Tip: Calculate the expect prize of binomial random variables ( including the expected value for multiple events ) using this online expected measure calculator .
Of course, calculating expected value ( EV ) gets more complicated in real life. For example, You buy one $ 10 raffle ticket for a new car valued at $ 15,000. Two thousand tickets are sold. What is the EV of your addition ? The rule for calculating the EV where there are multiple probabilities is :
E ( X ) = ΣX * P ( X )
Where Σ is summation notation .
The equality is basically the same, but here you are adding the sum of all the gains multiplied by their person probabilities rather of just one probability .
Other Expected Value Formulas
The two formulas above are the two most coarse forms of the expected value formulas that you ’ ll visualize in AP Statistics or elementary statistics. however, in more rigorous or advanced statistics classes ( like these ), you might come across the expected value formulas for continuous random variables or for the expected value of an arbitrary function.
Expected Value Formula for an arbitrary function
The expect value of a random varying is precisely the mean of the random variable star. You can calculate the EV of a continuous random varying using this formula :
Where f(x) is the
The “∫” symbol is called an
Where farad ( adam ) is the probability concentration function, which represents a officiate for the concentration curve.The “ ∫ ” symbol is called an builtin, and it is equivalent to finding the area under a arch If an event is represented by a affair of a random variable ( gravitational constant ( x ) ) then that routine is substituted into the EV for a continuous random variable convention to get :
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Watch the television for an example :
How to find an expect value Watch this television on YouTube
Can’t see the video? Can ’ t see the television ? Click here This section explains how to figure out the expect measure for a single item ( like purchasing a one raffle ticket ) and what to do if you have multiple items. If you have a discrete random variable, read Expected value for a discrete random variable .
Example question: You buy one $ 10 raffle tag for a new car valued at $ 15,000. Two thousand tickets are sold. What is the expect rate of your gain ?
Step 1: Make a probability chart (see: How to construct a probability distribution). Put Gain ( X ) and Probability P ( X ) heading the rows and Win/Lose heading the column.
Step 2: Figure out how much you could gain and lose. In our example, if we won, we ’ five hundred be up $ 15,000 ( less the $ 10 price of the raffle ticket ). If you lose, you ’ five hundred be devour $ 10. Fill in the datum ( I ’ meter using Excel here, so the damaging amounts are showing in red ) .
Step 3: In the bottom row, put your odds of winning or losing. Seeing as 2,000 tickets were sold, you have a 1/2000 gamble of winning. And you besides have a 1,999/2,000 probability opportunity of losing.
Read more: Coin MachineReal coin pusher
Step 4: Multiply the gains ( ten ) in the circus tent row by the Probabilities ( P ) in the penetrate quarrel.
$ 14,990 * 1/2000 = $ 7.495,
( – $ 10 ) * ( 1,999/2,000 ) = – $ 9.995
step 5 : Add the two values together :
$ 7.495 + – $ 9.995 = – $ 2.5 .
That ’ s it !
Note on multiple items : for example, what if you purchase a $ 10 ticket, 200 tickets are sold, and deoxyadenosine monophosphate well as a car, you have runner up prizes of a certificate of deposit musician and baggage set ?
Perform the steps precisely as above. Make a probability chart except you ’ ll have more items :
then multiply/add the probabilities as in tone 4 : 14,990* ( 1/200 ) + 100 * ( 1/200 ) + 200 * ( 1/200 ) + – $ 10 * ( 197/200 ) .
You ’ ll note nowadays that because you have 3 prizes, you have 3 chances of winning, so your chance of losing decreases to 197/200 .
Note on the formula: The actual formula for expected profit is E ( X ) =∑X*P ( X ) ( this is besides one of the AP Statistics convention ). What this is saying ( in English ) is “ The expect prize is the total of all the gains multiplied by their individual probabilities. ”
Like the explanation ? Check out the Practically Cheating Statistics Handbook, which has hundreds more bitbybit explanations, just like this one !
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step 1 : Type your values into two columns in Excel ( “ x ” in one column and “ farad ( x ) ” in the following.
mistreat 2 : Click an empty cellular telephone.
pace 3 : type =SUMPRODUCT ( A2 : A6, B2 : B6 ) into the cell where A2 : A6 is the actual placement of your ten variables and f ( ten ) is the actual location of your fluorine ( x ) variables.
step 4 : imperativeness Enter .
That ’ s it !
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You can think of an ask value as a entail, or average, for a probability distribution. A discrete random variable is a random variable that can merely take on a certain number of values. For exemplar, if you were rolling a die, it can lone have the typeset of numbers { 1,2,3,4,5,6 }. The expect respect formula for a discrete random variable star is :
Basically, all the recipe is telling you to do is find the mean by adding the probabilities. The mean and the expected value are so closely related they are basically the same thing. You ’ ll want to do this slenderly differently depending on if you have a located of values, a dress of probabilities, or a convention .
Expected Value Discrete Random Variable (given a list).
Example problem #1: The weights ( X ) of patients at a clinic ( in pounds ), are : 108, 110, 123, 134, 135, 145, 167, 187, 199. Assume one of the patients is chosen at random. What is the EV ?
step 1 : Find the mean. The mean is :
108 + 110 + 123 + 134 + 135 + 145 + 167 + 187 + 199 = 145.333.
That ’ s it !
Expected Value Discrete Random Variable (given “X”).
Example problem #2. You toss a bazaar mint three times. ten is the phone number of heads which appear. What is the EV ?
step 1 : figure out the potential values for X. For a three mint flip, you could get anywhere from 0 to 3 heads. then your values for X are 0, 1, 2 and 3 .
dance step 2 : trope out your probability of getting each value of X. You may need to use a sample space ( The sample space for this problem is : { HHH TTT TTH THT HTT HHT HTH THH } ). The probabilities are : 1/8 for 0 heads, 3/8 for 1 head, 3/8 for two heads, and 1/8 for 3 heads .
step 3 : Multiply your ten values in Step 1 by the probabilities from step 2.
E ( X ) = 0 ( 1/8 ) + 1 ( 3/8 ) + 2 ( 3/8 ) + 3 ( 1/8 ) = 3/2 .
The EV is 3/2 .
Expected Value Discrete Random Variable (given a formula, f(x)).
Example problem #3. You toss a mint until a buttocks comes up. The probability density affair is degree fahrenheit ( adam ) = ½x. What is the EV ?
footfall 1 : Insert your “ adam ” values into the beginning few values for the formula, one by one. For this particular formula, you ’ ll get :
1/20 + 1/21 + 1/22 + 1/23 + 1/24 + 1/25 .
step 2 : Add up the values from Step 1 :
= 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1.96875 .
Note: What you are looking for here is a issue that the series converges on ( i.e. a set number that the values are heading towards ). In this case, the values are headed towards 2, so that is your EV.
Tip : You can alone use the have a bun in the oven respect discrete random variable convention if your affair converges absolutely. In other words, the function must stop at a particular value. If it doesn ’ triiodothyronine converge, then there is no EV .
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Expected values for binomial random variables ( i.e. where you have two variables ) are probably the simplest type of expected values. In real life, you ’ re probably to encounter more complex expected values that have more than two possibilities. For example, you might buy a strike off lottery ticket with prizes of $ 1000, $ 10 and $ 1. You might want to know what the bribe is going to be if you go ahead and spend $ 1, $ 5 or tied $ 25 .
Let ’ s say your school is raffling off a season pas to a local theme park, and that value is $ 200. If the school sells one thousand $ 10 tickets, every person who buys the slate will lose $ 9.80, expect for the person who wins the season pass. That ’ s a fall back proposal for you ( although the school will rake it in ). You might want to save your money ! here ’ s the mathematics behind it :
 The value of winning the season ticket is $199 (you don’t get the $10 back that you spent on the ticket.
 The odds that you win the season pass are 1 out of 1000.
 Multiply (1) by (2) to get: $199 * 0.001 = 0.199. Set this number aside for a moment.
 The odds that you lose are 999 out of 1000. In other words, your odds of ending up minus ten dollars are 999/1000. Multiplying $10, you get 9.999.
 Adding (3) and (4) gives us the expected value: 0.199 + 9.999 = 9.80.
here ’ sulfur that scenario in a table :
What is the St. Petersburg Paradox?
The St. Petersburg Paradox has been stumping mathematicians for centuries. It ’ randomness about a dissipated game you can always win. But despite that fact, people aren ’ metric ton bequeath to pay a lot money to play it. It ’ s called the St. Petersburg Paradox because of where it appeared in print : in the 1738 Commentaries of the Imperial Academy of Science of Saint Petersburg .
The Paradox is this : There ’ s a simple count game you can play where your winnings are always going to be bigger than the amount of money you bet. Imagine buying a strike off lottery tag where the expected value ( i.e. the sum you can expect to win ) is always higher than the amount you pay for the ticket. You could buy a ticket for $ 1, $ 10, or a million dollars. You will always come up ahead. Would you play ?
Assuming the bet on international relations and security network ’ t rigged, you credibly should play. But the paradox is that most people wouldn ’ deoxythymidine monophosphate be willing to bet on a game like this for more than a few dollars. therefore, why is that ? There are a couple of potential explanations :
 People aren’t rational. They aren’t willing to risk their money even for a sure bet.
 There has to be something wrong with the game’s odds. Surely the odds of winning can’t always be that good, can they?
The short answer is, people are rational number ( for the most part ), they are bequeath to character with their money ( for the most part ). And, there is absolutely nothing improper with the game. If you ’ re confused at this point — that is why it ’ sulfur called a paradox .
The St. Petersburg Paradox Game.
n and the game would end. In other words, if tails come up on the first toss, you would win $21 = $2. If tails comes up on the third toss, you would win $23 = $8. And if you had a run and tails showed up on the 20th toss, you would win $220 = $1,048,576.
The master paradox wasn ’ thyroxine about lottery tickets ( they didn ’ t exist in 1738 ). It was about a coin flip game. Suppose you were asked by a supporter to play a coin flip game for $ 2. Assume the coin is bazaar ( i.e. it international relations and security network ’ metric ton weighted ). You toss the coin until the first tails comes up, at which time you would earn $ 2and the game would end. In other words, if tails come up on the first flip, you would win $ 2= $ 2. If tails comes up on the third toss, you would win $ 2= $ 8. And if you had a run and tails showed up on the twentieth pass, you would win $ 2= $ 1,048,576. If you figure out the expect value ( the expected bribe ) for this game, your electric potential winnings are infinite. For exemplar, on the foremost flip, you have a 50 % opportunity of winning $ 2. Plus you get to toss the coin again, so you besides have a 25 % luck of winning $ 4, plus a 12.5 % probability of winning $ 8 and so on. If you bet over and over again, your expected bribe ( gain ) is $ 1 each time you play, as shown by the follow postpone.
P(n)  Prize  Expected payoff 


1  1/2  $2  $1 
2  1/4  $4  $1 
3  1/8  $8  $1 
4  1/16  $16  $1 
5  1/32  $32  $1 
6  1/64  $64  $1 
7  1/128  $128  $1 
8  1/256  $256  $1 
9  1/512  $512  $1 
10  1/1024  $1024  $1 
You can ’ metric ton possibly lose money. still, despite the expect value being infinitely big, most people wouldn’t be willing to fork out more than a few bucks to play the game.
The St. Petersburg paradox has been debated by mathematicians for about three centuries. Why won ’ thyroxine people risk a set of money if the odds are surely in their favor ? As of so far, no one has found a satisfactory answer to the paradox. As Michael Clark states : “ [ The St. Petersburg Paradox ] seems to be one of those paradoxes which we have to swallow. ” A couple of solutions, which have been presented and yet have failed to offer a satisfactory answer :
 Limited utility (suggested by Jacob Bernoulli). Basically, the more we have of something, the less satisfied we are with it. You can apply this to candy; You’re likely to be satisfied with one bag, but after six or seven bags, you’re likely to not want any more. However, you can’t apply this to money. Everyone wants more money, right?
 Risk aversion. The average person might consider putting a few thousand dollars in the stock market. But they wouldn’t be willing to gamble their entire life savings. You can’t apply this rule to the St. Petersburg Paradox game because there is no risk.
following : Powerball Expected Value
References
Clark, Michael, 2002, “ The St. Petersburg Paradox ”, in Paradoxes from A to Z, London : Routledge, pp. 174–177.
Papoulis, A. “ Expected Value ; Dispersion ; Moments. ” §54 in Probability, Random Variables, and Stochastic Processes, 2nd erectile dysfunction. New York : McGrawHill, pp. 139152, 1984 .
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. “Expected Value in Statistics: Definition and Calculating it” From Stephanie Glen. “Expected Value in Statistics: Definition and Calculating it” From StatisticsHowTo.com : Elementary Statistics for the rest of us! https://ontopwiki.com/probabilityandstatistics/expectedvalue/
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