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: