FormulaBinomial random Variable

What is intended Value?

Expected worth is specifically what you could think it way intuitively: the return you can expect for some type of action, favor how numerous questions you might get appropriate if friend guess on a multiple choice test.

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For example, if you take it a 20 concern multiple-choice test through A,B,C,D as the answers, and you guess every “A”, climate you deserve to expect to obtain 25% best (5 the end of 20). The math behind this type of intended value is:The probability (P) of obtaining a question right if you guess: .25The number of questions top top the check (n)*: 20P x n = .25 x 20 = 5

*You can see this as X instead.

This form of expected value is dubbed an expected value for a binomial random variable. It’s a binomial experiment due to the fact that there are only two feasible outcomes: you gain the answer right, or you get the prize wrong.


Basic supposed Value Formula

The simple expected worth formula is the probability of an occasion multiplied by the lot of time the event happens:(P(x) * n).

The formula alters slightly follow to what type of events are happening. Because that most straightforward events, you’ll use either the intended Value formula the a Binomial random Variable or the expected Value formula because that Multiple Events.

Formula because that the meant Value that a Binomial arbitrarily Variable

The formula for the supposed Value for a binomial arbitrarily variable is: P(x) * X.X is the variety of trials and also P(x) is the probability of success. Because that example, if girlfriend toss a coin ten times, the probability of obtaining a top in each trial is 1/2 so the expected value (the variety of heads you deserve to expect to obtain in 10 coin tosses) is:P(x) * X = .5 * 10 = 5

Tip: calculate the intended value the binomial random variables (including the meant value because that multiple events) making use of this online expected value calculator.

Expected worth for many Events

Of course, calculating intended value (EV) it s okay more complicated in actual life. For example, you buy one $10 raffle ticket because that a brand-new car valued in ~ $15,000. 2 thousand tickets room sold. What is the EV of her gain? The formula for calculating the EV whereby there are multiple probabilities is:

E(X) = ΣX * P(X)Where Σ is summation notation.

The equation is usually the same, yet here friend are adding the sum of all the gains multiplied by their individual probabilities instead of just one probability.

Other intended Value Formulas

The two formulas over are the two most usual forms of the intended value formulas the you’ll watch in AP Statistics or primary school statistics. However, in more rigorous or advanced statistics great (like these), you can come throughout the expected value formulas for continuous random variables or because that the expected value of an arbitrarily function.Expected value Formula for an arbitrary Function

How much would you bet if friend could always win?

The original paradox wasn’t about lottery ticket (they didn’t exist in 1738). It was about a coin toss game. Mean you were asked by a girlfriend to play a coin toss video game for $2. Assume the coin is same (i.e. The isn’t weighted). Friend toss the coin till the very first tails comes up, in ~ which time you would earn $2n and also the game would end. In other words, if tails come up on the an initial toss, friend would win $21 = $2. If tails comes up on the third toss, friend would win $23 = $8. And if you had a run and tails showed up ~ above the 20th toss, you would success $220 = $1,048,576.

If you figure out the meant value (the meant payoff) for this game, your potential winnings space infinite. For example, ~ above the an initial flip, you have actually a 50% possibility of winning $2. To add you acquire to toss the coin again, so you likewise have a 25% chance of to win $4, add to a 12.5% opportunity of win $8 and so on. If you gambling over and also over again, your expected payoff (gain) is $1 each time friend play, as displayed by the complying with table.

P(n)PrizeExpected payoff

You can’t possibly shed money. Still, regardless of the meant value gift infinitely large, most civilization wouldn’t be ready to fork out more than a couple of bucks come play the game.

The St. Petersburg paradox has actually been discussed by mathematicians for nearly three centuries. Why won’t human being risk a the majority of money if the odds are certainly in your favor? together of yet, nobody has uncovered a satisfactory answer come the paradox. Together Michael Clark states: “ appears to be among those paradoxes which we have to swallow.” A couple of solutions, which have actually been presented and yet have failed to sell a satisfactory answer:

Risk aversion. The average human being might think about putting a couple of thousand dollars in the share market. However they wouldn’t be willing to gambling their entire life savings. Friend can’t use this dominance to the St. Petersburg Paradox game since there is no risk.


Clark, Michael, 2002, “The St. Petersburg Paradox”, in Paradoxes from A come Z, London: Routledge, pp. 174–177.Papoulis, A. “Expected Value; Dispersion; Moments.” §5-4 in Probability, random Variables, and also Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 139-152, 1984.

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