Expected value, also known as expectation, is a concept in mathematics that represents the average value or outcome of a random variable. It is a measure of the central tendency of a probability distribution.
The concept of expected value was first introduced by mathematician Jacob Bernoulli in the early 18th century. However, it was further developed and formalized by Pierre-Simon Laplace in the late 18th century. Since then, it has become an essential tool in probability theory and statistics.
Expected value is typically introduced in high school or college-level mathematics courses. It requires a solid understanding of probability and basic algebraic concepts.
Expected value involves the following key points:
Random Variables: A random variable is a numerical quantity that can take on different values based on the outcome of a random event.
Probability Distribution: A probability distribution describes the likelihood of each possible value of a random variable.
Weighted Average: Expected value is calculated as a weighted average of the possible values of a random variable, where the weights are determined by the probabilities associated with each value.
There are different types of expected value depending on the nature of the random variable:
Discrete Expected Value: This type is used when the random variable can only take on a finite or countable number of values.
Continuous Expected Value: This type is used when the random variable can take on any value within a certain range.
Expected value possesses several important properties:
Linearity: The expected value of a sum of random variables is equal to the sum of their individual expected values.
Independence: If two random variables are independent, the expected value of their product is equal to the product of their individual expected values.
To calculate the expected value, you need to follow these steps:
Identify the possible values of the random variable and their corresponding probabilities.
Multiply each value by its probability.
Sum up the products obtained in step 2.
The formula for expected value is as follows:
E(X) = x₁p₁ + x₂p₂ + x₃p₃ + ... + xn * pn
Where:
The expected value formula is applied in various scenarios, such as:
Gambling: It helps determine the average outcome of a game or bet.
Insurance: It assists in calculating the expected payout or premium for an insurance policy.
Decision Making: It aids in making rational decisions by considering the potential outcomes and their associated probabilities.
The symbol commonly used to represent expected value is E(X), where X is the random variable.
There are different methods to calculate expected value, including:
Probability Distribution: If the probability distribution of the random variable is known, the expected value can be calculated directly using the formula.
Frequency Distribution: If only the frequency distribution is given, the expected value can be estimated by dividing the sum of the products of values and frequencies by the total number of observations.
A fair six-sided die is rolled. What is the expected value of the outcome?
Solution: The possible outcomes are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6. E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5
A bag contains 5 red balls and 3 blue balls. If a ball is randomly drawn, what is the expected number of red balls?
Solution: The possible outcomes are 0 red balls (with probability 3/8) and 1 red ball (with probability 5/8). E(X) = (0 * 3/8) + (1 * 5/8) = 5/8
A company sells two products, A and B, with probabilities of 0.6 and 0.4, respectively. The profit from product A is $100 and from product B is $200. What is the expected profit?
Solution: E(X) = (100 * 0.6) + (200 * 0.4) = 160
A fair coin is tossed three times. What is the expected number of heads?
A bag contains 10 marbles, 4 red and 6 blue. If two marbles are randomly drawn without replacement, what is the expected number of red marbles?
A spinner is divided into three equal sections, labeled A, B, and C. The probabilities of landing on each section are 0.4, 0.3, and 0.3, respectively. If a player wins $10 for landing on section A and $5 for landing on section B, what is the expected winnings?
Q: What is the expected value (expectation)? A: Expected value is the average value or outcome of a random variable.
Q: How is expected value calculated? A: Expected value is calculated by multiplying each possible value of a random variable by its corresponding probability and summing up the products.
Q: What is the significance of expected value? A: Expected value helps in understanding the central tendency of a probability distribution and making informed decisions based on potential outcomes.
Q: Can expected value be negative? A: Yes, expected value can be negative if the random variable has negative values and their probabilities are significant.
Q: Is expected value the same as the most likely outcome? A: No, expected value represents the average outcome, which may not necessarily be the most likely outcome.