Final answer:
The expected value is the long-term average result of an experiment if it were repeated many times. To calculate it, multiply each possible outcome by its probability and sum these products. This calculation provides insight into what you can expect on average from random events over time.
Step-by-step explanation:
To find the expected value (E(X)) or mean (μ) of a discrete random variable X, you must calculate the sum of each possible value of the variable multiplied by its corresponding probability. The formula to compute the expected value is:
E(X) = μ = Σ xP(x)
For example, if you have a table of values for X alongside their probabilities P(x), you simply multiply each value of X (x) by its probability (P(x)) and sum all these products to get the expected value. This gives you a long-term average or mean if you were to repeat an experiment an infinite number of times.
Let's say the table shows two outcomes: losing $2 with probability 0.99999 and profiting $100,000 with probability 0.00001. The expected value would be calculated as:
E(X) = (-2)(0.99999) + (100000)(0.00001) = -1.99998 + 1 = -0.99998 ≈ -$1
The result indicates that, on average, you can expect to lose approximately $1 for each game played.