Please show work not just final answers Thanks 86 Roll two f

Please show work not just final answers!! Thanks.

86. Roll two fair dice. Each die has six faces. a. List the sample space. b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). C. Let B be the event that the sum of the two rolls is at most seven. Find P(B). d. In words, explain what P(A|B) represents. Find P(A|B). e. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification. f. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification.

Solution

a)

These are the possible results:

b)

Counting these events, there are 6 such events.

hence,

P(3 or 4, then even) = 6/36 = 1/6 [answer]

***********************

c)

Consider the table:

Here, there are 21 such events that the sum is at most 7. Hence,

P(at most 7) = 21/36 = 7/12 = 0.583333333 [answer]

*********************

d)

P(A|B) means \"the probability that a three or four occured first and it is followed by an even number, given that the sum is at most 7.\"

Among the 6 events of A, there are 3 of them with sum at most 7, namely, (3,2), (4,2), (3,4).

Thus,

P(B|A) = 3/6 = 0.5 [answer]

****************************

e)

They are not, because they have intersection (can happen together).

As we can see, those events that they happened together are (3,2), (4,2), (3,4).

P(A and B) = 3/36 = 1/12 =/= 0.

Thus, they are not mutually exclusive.

***********************

f)

They are not, as P(B}A) =/= P(B).

Die 1	Die 2
1	1
1	2
1	3
1	4
1	5
1	6
2	1
2	2
2	3
2	4
2	5
2	6
3	1
3	2
3	3
3	4
3	5
3	6
4	1
4	2
4	3
4	4
4	5
4	6
5	1
5	2
5	3
5	4
5	5
5	6
6	1
6	2
6	3
6	4
6	5
6	6