Question

If a pair of regular dice are tossed once use the expectation formula to determine the expected sum of the numbers on the upward faces of the two dice.

Answer 1

To find the expected sum of the numbers on the upward faces of the two dice, we need to first determine the possible outcomes and their respective probabilities.

When tossing two regular dice, each die has six possible outcomes: 1, 2, 3, 4, 5, or 6. Therefore, there are a total of 6 * 6 = 36 possible outcomes when tossing two dice.

To calculate the probability of each outcome, we can use the fact that the dice are regular, and thus, each outcome is equally likely. Therefore, the probability of each outcome is 1/36.

Now, we can calculate the expected value (E) using the expectation formula:

E = Σ(x * P(x))

where x is the value or outcome and P(x) is the probability of x.

Let's calculate the expected sum (E_sum):

E_sum = Σ(x * P(x))

E_sum = (2 * 1/36) + (3 * 2/36) + (4 * 3/36) + (5 * 4/36) + (6 * 5/36) + (7 * 6/36) + (8 * 5/36) + (9 * 4/36) + (10 * 3/36) + (11 * 2/36) + (12 * 1/36)

Simplifying the equation, we have:

E_sum = (2/36) + (6/36) + (12/36) + (20/36) + (30/36) + (42/36) + (40/36) + (36/36) + (30/36) + (22/36) + (12/36)

E_sum = 252/36

E_sum = 7

Therefore, the expected sum of the numbers on the upward faces of the two dice is 7.