Probability Distributions and Expected Value in Statistics

Probability Distributions

Constructing a Probability Distribution Table

A probability distribution table is a fundamental tool in statistics for representing the probabilities associated with each possible outcome of a random variable. It is especially useful for discrete random variables.

• Random Variable (X): A variable that takes on possible values from a random experiment.

• Probability (P(X)): The likelihood that the random variable takes a specific value.

To construct a probability distribution table:

1. List all possible outcomes for the random variable.

2. Assign probabilities to each outcome, ensuring the sum of all probabilities equals 1.

Example: Number of rotten tomatoes found in a sample.

Rotten Tomatoes X	0	1	2	3
Probability P(X)	0.95	0.02	0.02	0.01

Additional info: Probabilities are converted from percentages (e.g., 95% = 0.95).

Calculating the Mean (Expected Value) of a Probability Distribution

The expected value (mean) of a discrete probability distribution is the long-run average value of repetitions of the experiment it represents.

• Formula:

$E(X) = ext{mean} = extstyle extsum_{i} x_i P(x_i)$

• Multiply each value of X by its probability P(X).

• Add the results together.

Example Calculation:

• $0 imes 0.95 = 0$

• $1 imes 0.02 = 0.02$

• $2 imes 0.02 = 0.04$

• $3 imes 0.01 = 0.03$

Sum: $0 + 0.02 + 0.04 + 0.03 = 0.09$

Interpretation: On average, there are 0.09 rotten tomatoes per sample.

Expected Value in Decision Making

Expected Value in Raffles and Games of Chance

Expected value is a key concept in evaluating the fairness or profitability of games of chance, such as raffles.

• Gain (X): The amount you could win or lose.

• Probability (P(X)): The chance of each outcome occurring.

Step-by-Step Calculation

1. Construct a probability chart: List possible outcomes (win/lose), their associated gains/losses, and probabilities.

2. Determine gains/losses: For example, winning a raffle may yield $14,990 (after ticket cost), losing results in -$10.

3. Assign probabilities: If 2,000 tickets are sold, probability of winning is $1/2000$, losing is $1999/2000$.

4. Calculate expected value: Multiply each gain/loss by its probability and sum the results.

	Win	Lose
Gain X	$14,990	-$10
Probability P(X)	$1/2000$	$1999/2000$

Calculation:

• $14,990 imes rac{1}{2000} = 7.495$

• $-10 imes rac{1999}{2000} = -9.995$

Expected Value:

$E(X) = 7.495 + (-9.995) = -2.50$

Interpretation: On average, entering this raffle results in a loss of $2.50 per ticket. Thus, it is not a wise financial decision.

Key Terms and Concepts

• Probability Distribution: A table or function that shows the probabilities of outcomes for a random variable.

• Expected Value (Mean): The average outcome weighted by probability.

• Discrete Random Variable: A variable that can take on a countable number of values.

• Fair Game: A game where the expected value is zero.

Summary Table: Steps for Expected Value Calculation

Step	Description
1	List all possible outcomes and their probabilities.
2	Multiply each outcome by its probability.
3	Sum the products to get the expected value.

Additional info: These methods are foundational for analyzing risk and making decisions in statistics, economics, and finance.