Two fair dice, one blue and one red, are tossed, and the up face on each die is recorded. define the following events: e: { a 6 on the blue die }f: { the numbers are equal }find the following probabilities: (a) p(e)= (b) p(f)= (c) p(e? f)= (d) p(e|f)= (e) p(f|e)= are events e and f independent? a. yesb. no

a)

b)

c)

d)

e)

f) A. yes  

Step-by-step explanation:

We are assuming on this case that the two dice are fair. And we have the following events defined:

E:{ A 6 on the blue die }   F:{ The numbers are equal }

(a) P(E)=

For this case the probability of obtain a 6 on the blue die is

Since we have 6 possible optiosn and just one with a 6 on the blue die.

(b) P(F)=

First we need to define the sampling space when we tossed the two dice.

S={ (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)}

And as we can see we have just 6 pairs on which we have the same numbers for the two die. (1,1), (2,2),(3,3), (4,4),(5,5),(6,6), and the total outcomes are 36 so then:

(c) P(E and F)=

For this case we assume that the blue die is the first outcome on the sample space. We want to find on which pairs we have a 6 on the blue die and the numbers are equal in the two dice.

The only pair that satisfy this is (6.6) and we have a total of 36 possible outcomes so then:

(d) P(E|F)=

For this case we can use the following formula from the Bayes theorem:

(e) P(F|E)=

For this case we can use the following formula from the Bayes theorem:

Are events E and F independent?A. yesB. no

A. yes

Because we have that the following condition satisfied:

7. d. 64

8. i. 32

9. c. 125

10. j. none of the above

11. d. 81

12. i. 216

13. e. none of the above

14. g. 240

15. e. none of the above

16. g. 32

17.   e. none of the above