# Expected Value in Statistics: Definition and Calculating it

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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 examination .
Watch this television for a quick explanation of the expected value formula :

Can’t see the video? Can ’ metric ton see the video ? Click here For model, if you take a 20 question multiple-choice examination with A, B, C, D as the answers, and you guess all “ A ”, then you can expect to get 25 % correct ( 5 out of 20 ). The math behind this kind of expected value is :
The probability ( P ) of getting a interview right if you guess : .25
The count of questions on the quiz ( newton ) * : 20
P x n = .25 x 20 = 5
*You might see this as X rather .
This type of expected value is called an expected value for a binomial random variable. It ’ s a binomial experiment because there are lone two possible outcomes : you get the suffice right, or you get the answer wrong .
The basic expected value rule is the probability of an event multiplied by the amount of times the event happens:
(P(x) * n)
.
The rule changes slenderly according to what kinds of events are happening. For most simple events, you ’ ll function either the Expected Value recipe 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.
x is the numeral of trials and P ( x ) is the probability of success. For model, if you toss a mint ten times, the probability of getting a heads in each test is 1/2 so the expected value ( the number of heads you can expect to get in 10 mint tosses ) is :
P ( x ) * X = .5 * 10 = 5
Tip: Calculate the ask value of binomial random variables ( including the expected value for multiple events ) using this on-line expected rate calculator .
Of naturally, calculating expected value ( EV ) gets more complicated in very life. For exercise, You buy one \$ 10 raffle tag for a newly car valued at \$ 15,000. Two thousand tickets are sold. What is the EV of your reach ? The recipe for calculating the EV where there are multiple probabilities is :
E ( X ) = ΣX * P ( X )
Where Σ is sum notation .
The equation is basically the same, but here you are adding the sum of all the gains multiplied by their person probabilities alternatively of fair one probability .

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### Other Expected Value Formulas

The two formulas above are the two most coarse forms of the expected value formulas that you ’ ll see 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 serve
The ask value of a random variable is just the mean of the random variable. You can calculate the EV of a continuous random variable using this formula :

Where f(x) is the
The “∫” symbol is called an
Where degree fahrenheit ( x ) is the probability density function, which represents a function for the density curve.The “ ∫ ” symbol is called an integral, and it is equivalent to finding the area under a curve If an event is represented by a function of a random varying ( guanine ( x ) ) then that officiate is substituted into the EV for a continuous random variable recipe to get :

Watch the video for an model :

Can’t see the video? Can ’ triiodothyronine see the television ? Click hera This department explains how to figure out the expect value for a unmarried item ( like purchasing a single raffle tag ) 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 ticket for a modern car valued at \$ 15,000. Two thousand tickets are sold. What is the expect value 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 a lot you could gain and lose. In our case, 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 down \$ 10. Fill in the datum ( I ’ megabyte using Excel here, so the negative amounts are showing in bolshevik ) . Step 3: In the bed rowing, put your odds of winning or losing. Seeing as 2,000 tickets were sold, you have a 1/2000 chance of winning. And you besides have a 1,999/2,000 probability find of losing . Step 4: Multiply the gains ( ten ) in the crown row by the Probabilities ( P ) in the bed row.
\$ 14,990 * 1/2000 = \$ 7.495,
( – \$ 10 ) * ( 1,999/2,000 ) = – \$ 9.995
dance step 5 : Add the two values together :
\$ 7.495 + – \$ 9.995 = – \$ 2.5 .
That ’ s it !

Note on multiple items : for case, what if you purchase a \$ 10 tag, 200 tickets are sold, and deoxyadenosine monophosphate well as a cable car, you have runner up prizes of a cadmium player and baggage set ?

Perform the steps precisely as above. Make a probability graph except you ’ ll have more items : then multiply/add the probabilities as in step 4 : 14,990* ( 1/200 ) + 100 * ( 1/200 ) + 200 * ( 1/200 ) + – \$ 10 * ( 197/200 ) .
You ’ ll note now that because you have 3 prizes, you have 3 chances of winning, so your opportunity of losing decreases to 197/200 .
Note on the formula: The actual convention for expected gain is E ( X ) =∑X*P ( X ) ( this is besides one of the AP Statistics rule ). What this is saying ( in English ) is “ The expect measure is the total of all the gains multiplied by their person probabilities. ”

Like the explanation ? Check out the Practically Cheating Statistics Handbook, which has hundreds more bit-by-bit explanations, just like this one ! step 1 : Type your values into two columns in Excel ( “ x ” in one column and “ f ( x ) ” in the next.
gradation 2 : Click an empty cell.
pace 3 : type =SUMPRODUCT ( A2 : A6, B2 : B6 ) into the cellular telephone where A2 : A6 is the actual placement of your ten variables and farad ( adam ) is the actual location of your fluorine ( x ) variables.
step 4 : imperativeness Enter .
That ’ s it !
You can think of an expected value as a base, or median, for a probability distribution. A discrete random variable is a random variable that can lone take on a certain number of values. For exercise, if you were rolling a die, it can only have the set of numbers { 1,2,3,4,5,6 }. The expect value rule for a discrete random variable is : Basically, all the formula is telling you to do is find the bastardly by adding the probabilities. The mean and the expected value are so closely related they are basically the same thing. You ’ ll necessitate to do this slenderly differently depending on if you have a set of values, a adjust 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 bastardly 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 fairly mint three times. x is the total of heads which appear. What is the EV ?
footprint 1 : figure out the possible values for X. For a three mint toss, you could get anywhere from 0 to 3 heads. so your values for X are 0, 1, 2 and 3 .
step 2 : name 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 coin until a chase comes up. The probability density function is f ( adam ) = ½x. What is the EV ?
footfall 1 : Insert your “ x ” values into the beginning few values for the rule, one by one. For this particular formula, you ’ ll contract :
1/20 + 1/21 + 1/22 + 1/23 + 1/24 + 1/25 .
mistreat 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 hera is a number that the series converges on ( i.e. a set number that the values are heading towards ). In this lawsuit, the values are headed towards 2, so that is your EV.
Tip : You can only use the expect value discrete random variable formula if your affair converges absolutely. In early words, the function must stop at a particular value. If it doesn ’ thyroxine converge, then there is no EV .
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 likely to encounter more complex expected values that have more than two possibilities. For example, you might buy a start off lottery tag with prizes of \$ 1000, \$ 10 and \$ 1. You might want to know what the return is going to be if you go ahead and spend \$ 1, \$ 5 or even \$ 25 .
Let ’ s say your school is raffling off a season pass to a local theme park, and that rate is \$ 200. If the school sells one thousand \$ 10 tickets, every person who buys the ticket will lose \$ 9.80, expect for the person who wins the season elapse. That ’ s a suffer suggestion for you ( although the school will rake it in ). You might want to save your money ! here ’ s the mathematics behind it :

1. The value of winning the season ticket is \$199 (you don’t get the \$10 back that you spent on the ticket.
2. The odds that you win the season pass are 1 out of 1000.
3. Multiply (1) by (2) to get: \$199 * 0.001 = 0.199. Set this number aside for a moment.
4. 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.
5. 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 ’ s about a dissipated bet on you can always win. But despite that fact, people aren ’ thymine will to pay much money to play it. It ’ sulfur 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 bet game you can play where your winnings are always going to be bigger than the measure of money you bet. Imagine buying a chicken feed off lottery ticket where the expected value ( i.e. the amount you can expect to win ) is always higher than the sum 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 game international relations and security network ’ triiodothyronine rigged, you probably should play. But the paradox is that most people wouldn ’ metric ton be volition to bet on a game like this for more than a few dollars. so, why is that ? There are a pair of potential explanations :

1. People aren’t rational. They aren’t willing to risk their money even for a sure bet.
2. 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 ( for the most separate ), they are will to part with their money ( for the most share ). And, there is absolutely nothing wrong with the game. If you ’ re confused at this point — that is why it ’ second 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 original paradox wasn ’ t about lottery tickets ( they didn ’ t exist in 1738 ). It was about a mint convulse crippled. Suppose you were asked by a friend to play a mint convulse game for \$ 2. Assume the coin is fair ( i.e. it international relations and security network ’ metric ton weighted ). You toss the mint until the first tails comes up, at which prison term you would earn \$ 2and the bet on would end. In other words, if tails come up on the foremost discard, you would win \$ 2= \$ 2. If tails comes up on the third flip, you would win \$ 2= \$ 8. And if you had a run and tails showed up on the twentieth flip, you would win \$ 2= \$ 1,048,576. If you figure out the expect value ( the expected return ) for this game, your potential winnings are infinite. For exercise, on the first flip, you have a 50 % chance of winning \$ 2. Plus you get to toss the coin again, so you besides have a 25 % opportunity of winning \$ 4, plus a 12.5 % casual of winning \$ 8 and so on. If you bet over and complete again, your expected payoff ( derive ) is \$ 1 each clock you play, as shown by the play along table.

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 ’ triiodothyronine possibly lose money. hush, despite the ask 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 ’ triiodothyronine people risk a fortune of money if the odds are surely in their prefer ? As of even, 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 so far 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.

## References

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