Expected Value

Combinatorics Progress

Completed0/11

0 completed0 in progress

Expected Value

What expected value is and the different types of questions you could get on a contest.

Author: Jaden Park

Introduction

The expected value (EV), also known as the expectation, the average, or the mean value, is a fundamental concept in probability theory and statistics. It gives us a measure of the center of a probability distribution and represents the average outcome we would expect if we were to repeat an experiment an infinite number of times.

Definition

Mathematically, the expected value of a discrete random variable $X$ is defined as:

E(X) = \sum_{i=1}^{n} x_i p_i

where $x_i$ represents each possible value the random variable can take, and $p_i$ is the probability of $x_i$ .

For a continuous random variable, the expected value is defined using an integral:

E(X) = \int_{-\infty}^{\infty} x f(x) dx

where $f(x)$ is the probability density function of $X$ .

Here are some examples.

Example (Discrete Random Variable)

Consider a simple dice game where you win 4 dollars if you roll a 6, and lose 1 dollar if you roll any other number. What is the expected value of this game?

Solution

The game has two possible outcomes:

Rolling a $6:$ You win $4$ dollars, with a probability of $\frac{1}{6}$ .
Rolling any other number $(1, 2, 3, 4,$ or $5):$ You lose $1$ dollar, with a probability of $\frac{5}{6}$ .

The expected value (EV) is calculated as:

\text{EV} = (4 \cdot \frac{1}{6}) + (-1 \cdot \frac{5}{6}) = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6}

So, the expected value of playing this game is $-\frac{1}{6}$ dollars, meaning on average, you expect to lose approximately $0.17$ dollars every time you play this game.

Example (Continuous Random Variable)

Imagine an insurance company charges $100$ dollars for a policy that pays $1,000$ dollars in the event of a certain accident, which has a $0.1$ probability of happening within the policy period. What is the expected profit or loss for the insurance company per policy sold?

Solution

The insurance company's profit or loss depends on whether the accident happens:

No accident: The company keeps the $100$ dollars paid for the policy, with a probability of $0.9$ (since the accident has a $0.1$ probability).
Accident occurs: The company pays out $1,000$ dollars but has already received $100$ dollars for the policy, resulting in a net loss of $900$ dollars, with a probability of $0.1$ .

The expected value (EV) is calculated as:

\text{EV} = (100 \cdot 0.9) + (-900 \cdot 0.1) = 90 - 90 = 0

The expected value is $0$ dollars, meaning that, on average, the insurance company neither makes a profit nor a loss per policy sold, before considering operational costs.

Here are some practice problems that are harder.

Practice Problem

(2010 HMMT Guts Q18)

Jeff has a $50$ point quiz at $11$ am. He wakes up at a random time between $10$ am and noon, then arrives at class $15$ minutes later. If he arrives on time, he will get a perfect score, but if he arrives more than $30$ minutes after the quiz starts, he will get a $0,$ but otherwise, he loses a point for each minute he’s late (he can lose parts of one point if he arrives a nonintegral number of minutes late). What is Jeff’s expected score on the quiz?

Practice Problem

(2009 HMMT General Part 1 Q5)

Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

Final Notes and Tips

Not super common to see, would still understand the idea behind it.

Pascal's Triangle

Geometric Probability

Combinatorics Progress

Counting

Principles

Miscellaneous

Introduction

Definition

Example (Discrete Random Variable)

Solution

Example (Continuous Random Variable)

Solution

Practice Problem

Practice Problem

Final Notes and Tips

Introduction

Definition

Example (Discrete Random Variable)

Solution

Example (Continuous Random Variable)

Solution

Practice Problem

Practice Problem

Final Notes and Tips