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.
The expectation formula states that for each possible outcome, we multiply the outcome by its probability and sum them up.
In this case, we have two regular dice, each having 6 sides numbered from 1 to 6. To find the expected sum of the numbers on the upward face of the two dice, we need to consider all possible outcomes.
There are 36 total possible outcomes when two dice are tossed (6 outcomes for the first die and 6 outcomes for the second die). Since each outcome is equally likely, the probability for each outcome is 1/36.
To find the expected sum, we sum up the products of the outcomes and their respective probabilities:
Expected sum = (1 * 1/36) + (2 * 1/36) + (3 * 1/36) + ... + (12 * 1/36)
Simplifying this expression, we get:
Expected sum = 1/36 + 2/36 + 3/36 + ... + 12/36
To find the sum of the series 1/36 + 2/36 + 3/36 + ... + 12/36, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
In this case, the first term is 1/36 and the last term is 12/36, so the number of terms (n) is 12.
Sum = (12/2)(1/36 + 12/36)
Sum = 6(13/36)
Sum = 13/6
Therefore, the expected sum of the numbers on the upward faces of the two dice is 13/6 or approximately 2.17.
To determine the expected sum of the numbers on the upward faces of two regular dice, we first need to find the probability of each possible sum.
There are 6 possible outcomes for each die, ranging from 1 to 6. Since there are two dice, the total number of outcomes is 6 * 6 = 36.
To find the probability of each sum, we create a table and count how many ways we can achieve each possible sum:
Sum | Ways to Achieve | Probability
-----------------------------------
2 | 1 | 1/36
3 | 2 | 2/36
4 | 3 | 3/36
5 | 4 | 4/36
6 | 5 | 5/36
7 | 6 | 6/36
8 | 5 | 5/36
9 | 4 | 4/36
10 | 3 | 3/36
11 | 2 | 2/36
12 | 1 | 1/36
Now, we can calculate the expected value using the formula:
Expected Value = Sum of (Probability * Value) for each sum
Expected Value = (1/36 * 2) + (2/36 * 3) + (3/36 * 4) + (4/36 * 5) + (5/36 * 6) + (6/36 * 7) + (5/36 * 8) + (4/36 * 9) + (3/36 * 10) + (2/36 * 11) + (1/36 * 12)
Simplifying the expression:
Expected Value = 2/36 + 6/36 + 12/36 + 20/36 + 30/36 + 42/36 + 40/36 + 36/36 + 30/36 + 22/36 + 12/36
Expected Value = 252/36
Simplifying the fraction:
Expected Value = 7
Therefore, the expected sum of the numbers on the upward faces of the two dice is 7.