The table shows the probability distribution of a fair six-sided die. Determine 𝐸 of 𝑥.
The top row of our table, labelled 𝑥, shows all the possible outcomes on the die, numbers one, two, three, four, five, and six. The bottom row shows the probability of 𝑥 equals 𝑥. The probability that we land on a one when rolling a fair six-sided die is one-sixth. Likewise, the probability of landing on a two, a three, a four, a five, or a six are all equal to one-sixth.
We’ve been asked to determine or calculate 𝐸 of 𝑥, which is the mean or the expected value. This expected value is calculated by working out the sum of 𝑥 multiplied by the probability that 𝑥 equals 𝑥. Firstly, we multiply one by one-sixth. Then we add two multiplied by one-sixth.
Continuing for the other values in our table gives us three multiplied by one-sixth, four multiplied by one-sixth, five multiplied by one-sixth, and finally six multiplied by one-sixth. We could type this calculation into the calculator in one go.
Alternatively, each of the individual multiplication sums give us one-sixth, one-third, one-half, two-thirds, five-sixths, and one. Finding the sum of these numbers gives us seven-halves, which is the same as 3.5. Therefore, the 𝐸 of 𝑥, the expected value or mean, is 3.5.