Could Someone Please Help Me Answer This Statistics Problem?

Question

The Question: For this assignment, you will create and use a probability distribution that reflects rolling a pair of dice. First, construct a probability distribution for rolling 5's, displaying it as a table. Be sure to include a column for x and a column for P(x). The P(x) numbers should be presented in decimal form, and must add up to 1. Next, use the P(x) values to calculate the mean, standard deviation, and variance for x (Remember...x = number of fives rolled when two fair dice are tossed at the same time not a total of 5!).

I have determined that the probabilities of getting 5's when rolling a pair of dice is 0, 1, and 2, seeing as you the possibilities can either be that you roll zero 5's, you roll one 5, or you roll two 5's. I just need help calculating the probabilities.

The table should look similar to below:

x P(x)

------------------

0 ?

1 ?

2 ?

Answer 1

When rolling a pair of dice, the possible outcomes for each die are 1, 2, 3, 4, 5, and 6. The probability of rolling a 5 on a single die is 1/6.

Let’s denote (X) as the number of fives rolled when two fair dice are tossed at the same time. The possible values for (X) are 0, 1, and 2.

1. (X = 0): This means that neither of the dice shows a 5. The probability of not rolling a 5 on a single die is 5/6. Since the two dice are independent, the probability of not rolling a 5 on either die is ((5/6) \times (5/6) = 25/36).
2. (X = 1): This means that exactly one of the dice shows a 5. There are two ways this can happen: either the first die shows a 5 and the second does not, or the first die does not show a 5 and the second does. The probability of either of these events is ((1/6) \times (5/6) = 5/36). Since there are two such events, the total probability is (2 \times (5/36) = 10/36).
3. (X = 2): This means that both dice show a 5. The probability of this event is ((1/6) \times (1/6) = 1/36).

So, the probability distribution is:

X
0	25/36
1	10/36
2	1/36

To calculate the mean, standard deviation, and variance, we use the following formulas:

1. Mean ((\mu)): (\mu = \sum x \cdot P(x)) = 0*25/36 + 1*10/36 + 2*1/36 = 1/3

Variance ((\sigma^2)): (\sigma^2 = \sum (x - \mu)^2 \cdot P(x))= (0-1/3)² *25/36 + (1-1/3)² *10/36 + (2-1/3)² *1/36 = 0.2778

1. Standard Deviation ((\sigma)): (\sigma = \sqrt{\sigma^2})

Substitute the values of (X) and (P(X)) into these formulas to get the mean, variance, and standard deviation.