Question

In: Statistics and Probability

2. Three fair dice are rolled. Let X be the sum of the 3 dice. (a)...

2. Three fair dice are rolled. Let X be the sum of the 3 dice.

(a) What is the range of values that X can have?

(b) Find the probabilities of the values occuring in part (a); that is, P(X = k) for each k in part (a). (Make a table.)

3. Let X denote the difference between the number of heads and the number of tails obtained when a coin is tossed n times.

(a) What are the possible values of X?

(b) Suppose that the coin is fair, and that n = 3. What are the probabilities associated with each of the values that X can take?

7. An urn holds 10 red and 6 green marbles. A fair coin is tossed. If we get heads, two marbles are selected from the urn. Otherwise three marbles are selected. If we have only reds, what is the probability that we selected exactly two marbles?

Solutions

Expert Solution

Answer 2

3 fair die are rolled. Let X be the sum of the 3 dice.

The range of values that X can take will be:

Minimum sum: 1+1+1= 3

Maximum value: 6+6+6=18

Thus, the range of values are 3 to 18

Now it is important to understand that just as one die has six outcomes and two dice have 6² = 36 outcomes, rolling three dice will have 6³ = 216 outcomes.

There is one possible way three dice can total 3

3 ways for 4

6 for 5

10 for 6

15 for 7

21 for 8

25 for 9

27 for 10

27 for 11

25 for 12

21 for 13

15 for 14

10 for 15

6 for 16

3 for 17

1 for 18

Probability of a sum of 3: 1/216 = 0.5%

Probability of a sum of 4: 3/216 = 1.4%

Probability of a sum of 5: 6/216 = 2.8%

Probability of a sum of 6: 10/216 = 4.6%

Probability of a sum of 7: 15/216 = 7.0%

Probability of a sum of 8: 21/216 = 9.7%

Probability of a sum of 9: 25/216 = 11.6%

Probability of a sum of 10: 27/216 = 12.5%

Probability of a sum of 11: 27/216 = 12.5%

Probability of a sum of 12: 25/216 = 11.6%

Probability of a sum of 13: 21/216 = 9.7%

Probability of a sum of 14: 15/216 = 7.0%

Probability of a sum of 15: 10/216 = 4.6%

Probability of a sum of 16: 6/216 = 2.8%

Probability of a sum of 17: 3/216 = 1.4%

Probability of a sum of 18: 1/216 = 0.5%

Thus, the table is:

3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
0.5%	1.4%	2.8%	4.6%	7.0%	9.7%	11.6%	12.5%	12.5%	11.6%	9.7%	7.0%	4.6%	2.8%	1.4%	0.5%

Question 2 solved. Please post question 3 and question 7 as separate questions.

Happy learning!

orchestra answered 10 months ago

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