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.
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.
To determine the expected sum of the numbers on the upward faces of two dice, we can use the expectation formula.
The expectation formula states that the expected value (E) of a random variable X is the sum of the products of each possible outcome (x) and their respective probabilities (P(x)).
In this case, the random variable X represents the sum of the numbers on the upward faces of two dice.
Step 1: Determine the possible outcomes and their probabilities:
When rolling a pair of dice, the numbers on their upward faces range from 1 to 6. The sum of the numbers on the two dice can range from 2 to 12.
The possible outcomes and their probabilities are as follows:
Sum of 2: Probability = 1/36 (There is only one way to roll a sum of 2: rolling a 1 on both dice.)
Sum of 3: Probability = 2/36 (There are two ways to roll a sum of 3: rolling a 1 and a 2, or rolling a 2 and a 1.)
Sum of 4: Probability = 3/36 (There are three ways to roll a sum of 4: rolling a 1 and a 3, rolling a 2 and a 2, or rolling a 3 and a 1.)
Sum of 5: Probability = 4/36 (There are four ways to roll a sum of 5: rolling a 1 and a 4, rolling a 2 and a 3, rolling a 3 and a 2, or rolling a 4 and a 1.)
Sum of 6: Probability = 5/36 (There are five ways to roll a sum of 6: rolling a 1 and a 5, rolling a 2 and a 4, rolling a 3 and a 3, rolling a 4 and a 2, or rolling a 5 and a 1.)
Sum of 7: Probability = 6/36 (There are six ways to roll a sum of 7: rolling a 1 and a 6, rolling a 2 and a 5, rolling a 3 and a 4, rolling a 4 and a 3, rolling a 5 and a 2, or rolling a 6 and a 1.)
Sum of 8: Probability = 5/36
Sum of 9: Probability = 4/36
Sum of 10: Probability = 3/36
Sum of 11: Probability = 2/36
Sum of 12: Probability = 1/36
Step 2: Calculate the expected value using the expectation formula:
E(X) = (2 * P(sum of 2)) + (3 * P(sum of 3)) + (4 * P(sum of 4)) + (5 * P(sum of 5)) + (6 * P(sum of 6)) + (7 * P(sum of 7))
+ (8 * P(sum of 8)) + (9 * P(sum of 9)) + (10 * P(sum of 10)) + (11 * P(sum of 11)) + (12 * P(sum of 12))
E(X) = (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)
Step 3: Simplify the equation and calculate the expected value:
E(X) = (2/36) + (6/36) + (12/36) + (20/36) + (30/36) + (42/36) + (40/36) + (36/36) + (30/36) + (22/36) + (12/36)
E(X) = 252/36
E(X) = 7
Therefore, the expected sum of the numbers on the upward faces of the two dice is 7.