Provide one example to show how you can use the Expected Value computation to assess the fairness of a situation (probability experiment). Provide the detailed steps and calculations.
Sure! The concept of expected value can be used to assess the fairness of a situation or probability experiment. Let's take a simple example of rolling a fair six-sided die.
Step 1: Define the event and its associated outcomes:
In this case, the event is rolling the die, and the possible outcomes are the numbers 1, 2, 3, 4, 5, or 6.
Step 2: Assign probabilities to each outcome:
Since the die is fair, each outcome has an equal probability of occurring. Therefore, the probability of rolling any particular number is 1/6.
Step 3: Calculate the expected value:
The expected value is calculated by multiplying each outcome by its corresponding probability and summing them up.
Expected Value = (Outcome 1 * Probability 1) + (Outcome 2 * Probability 2) + ... + (Outcome n * Probability n)
Expected Value = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)
Expected Value = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
Expected Value = 21/6
Step 4: Simplify the expected value:
21/6 can be simplified as 3.5.
Therefore, the expected value of rolling a fair six-sided die is 3.5.
Step 5: Assess the fairness of the situation:
In this case, the expected value of 3.5 represents the average outcome we can expect over a large number of rolls. If a player were to bet on the outcome of rolling the die, and the player were to receive a reward equal to the number rolled, then over a large number of rolls, the player would expect to break even, as the average outcome is 3.5.
If the reward offered for rolling the die were less than 3.5, it would be an unfair game for the player. Conversely, if the reward were greater than or equal to 3.5, it would be a fair or favorable game for the player.
So, using the expected value computation, we can assess the fairness of the situation in terms of probabilities and outcomes.