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! Let's consider a simple example of flipping a fair coin. In this case, there are two possible outcomes: heads (H) or tails (T).
Step 1: Define the event and its outcomes
Let's define the event as "getting heads" and the outcomes as {H, T}.
Step 2: Assign probabilities to the outcomes
Since the coin is fair, the probability of getting heads (H) or tails (T) is each 1/2 or 0.5.
Step 3: Define the random variable
The random variable (X) represents the outcome of the event "getting heads." We assign the value 1 to getting heads (H) and 0 to getting tails (T). So, X = 1 if heads (H) and X = 0 if tails (T).
Step 4: Calculate the expected value
The expected value of a random variable (E[X]) can be calculated using the formula:
E[X] = (X1 * P1) + (X2 * P2) + ... + (Xn * Pn)
In our case, there are two possible outcomes: heads (H) and tails (T).
E[X] = (1 * 0.5) + (0 * 0.5)
E[X] = 0.5
Step 5: Interpretation
The expected value (0.5) represents the average outcome if we repeated the coin flip experiment many times. In this case, if we conducted a large number of flips, we can expect to get heads approximately 50% of the time.
Step 6: Assessing fairness
We can use the expected value to assess the fairness of the situation. Since the probability of getting heads is 0.5, the expected value of 0.5 indicates that, on average, half of the flips will result in heads. This suggests that the coin flip is fair, as there is an equal chance of getting either outcome.
In summary, by using the expected value calculation, we can assess the fairness of a situation – in this case, a coin flip – by comparing the calculated expected value to the probabilities assigned to the different outcomes.