## 1. Definitions

### 1.1. Mathematical expectation, or expected value, of a random variable X which can take on values a1, a2, …, an, with probabilities p1, p2, …, pn, respectively is defined as E(X) = a1 p1 + a2 p2 + … + an pn

1.1.1. Expected Value does NOT need to be a member of the sample space

### 1.2. For binomial distribution, with a fixed amount of n trials, each with p probability of success, the expected value is np.

1.2.1. Expected Value does NOT need to be a member of the sample space

## 2. Related topics

### 2.1. Weighted Averages

2.1.1. Ex: A math class is graded on 3 tests, each 20% of the final grade, and a final exam which is 40% of the final grade. If a student averaged 89% on the first 3 tests, what score is needed on the final exam to average a 90% for the class?

### 2.2. Binomial Distributions

2.2.1. Ex: A family has 3 children. If the possible gender of each child is 50/50 M/F, what is the probability the family has 0,1,2, or 3 girls?

## 3. Applications

### 3.1. Games of chance

3.1.1. Gambling

3.1.1.1. Roulette

3.1.1.2. Blackjack

3.1.1.3. Poker

3.1.2. Family Games

3.1.2.1. Yahtzee

3.1.2.2. Pig

### 3.2. Insurance

3.2.1. Risk Analysis

### 3.3. Medical Field

3.3.1. Drug Testing