Expected value

Given an event with a variety of different possible outcomes, the expected value is what one should expect to be the average outcome if the event were to be repeated many times. Note that this is not the same as the "most likely outcome."

Expected Value Video

A video that goes over the type of Expected value, practical examples, and problems: https://youtu.be/TCFoRx2R2ew

Video Example

https://www.youtube.com/watch?v=BBD66Q3KXuI

Formal Definition

More formally, we can define expected value as follows: if we have an event $Z$ whose outcomes have a discrete probability distribution, the expected value $E(Z) = \sum_z P(z) \cdot z$ where the sum is over all outcomes $z$ and $P(z)$ is the probability of that particular outcome. If the event $Z$ has a continuous probability distribution, then $E(Z) = \int_z P(z)\cdot z\ dz$ .

Properties

For any random variable $X$ and constant $c$ , $\text{E}[X+c]=\text{E}[X]+c.$
(Linearity of Expectation) For any random variables $X$ and $Y$ , $\text{E}[X+Y]=\text{E}[X]+\text{E}[Y].$
For any random variable $X$ and constant $c$ $\text{E}[cX]=c\text{E}[X].$

Uses

Expected value can be used to, for example, determine the price for playing a probability based carnival game. You first find the expected value per player. Then you can charge a reasonable price so that you gain money, but the price isn't unreasonably high.
Expected value can be used to calculate winning strategies in games of chance, such as board games.

Example

As an example, flipping a fair coin has two possible outcomes, heads (denoted here by $H$ ) or tails ( $T$ ). If we flip a fair coin repeatedly, we expect that we will get about the same number of heads as tails, or half as many as the total number of flips. Thus, the average outcome is $\frac 12 H + \frac 12 T$ . Note that not only is this not the most likely outcome, it is not even a possible outcome for a single flip.

Problems

Introductory

Find the expected value of a dice roll.
- Find the expected value of a weighted dice roll, where each dot has equal probability of being on top.

Intermediate

In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

(Source)

An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. The case $n = 11$ is shown here, #3. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with V as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n$ .

(USAMTS year 17, round 2, problem 3)

We play a game. The pot starts at $$0$ . On every turn, you flip a fair coin. If you flip heads, I add $$100$ to the pot. If you flip tails, I take all of the money out of the pot, and you are assessed a “strike.” You can stop the game before any flip and collect the contents of the pot, but if you get 3 strikes, the game is over and you win nothing. Find, with proof, the expected value of your winnings if you follow an optimal strategy.

(USAMTS year 17, round 4, problem 3)

Olympiad

Let $p_{n}(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^{n}k p_{n}(k)=n!$ .

(IMO 1987, #1)

Prove that there exists a 4-coloring of the set $M=\{1,2,3,\cdots,1987\}$ such that any 10-term arithmetic progression in the set $M$ is not monochromatic.

(IMO Shortlist, 1987)

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