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Two distinguishable (so order matters) fair dice are rolled. Define the following events: A = {sum of the numbers rolled is odd} B = {Sum of the numbers rolled is greater than 8} (a) Find the probabilities P(A) and P(B). First define events A and B (in set notation) to list out all outcomes in each event. (b) Find the probabilities P(A or B) and P(A and B). First define the events "A or B" and "A and B" explicitly by writing out the sample spaces. (c) Are the events A and B independent? Why or why not?

Respuesta :

Answer:

(a)P(A)=1/2, P(B)=5/18

(b)P(A and B)=1/6

P(A or B)=11/18

(c)A and B are not independent events.

Step-by-step explanation:

The sample space of rolling two dice is given as:

(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).

n(S)=36

Event A=The sum of the numbers rolled is odd

Sample Space of Event A

={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),

(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),

(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)}

n(A)=18

Event B= {Sum of the numbers rolled is greater than 8}

Sample Space of B={

{(3,6).

(4,5),(4,6).

(5,4),(5,5),(5,6).

(6,3),(6,4),(6,5),(6,6).}

n(B)=10

(a)P(A)=n(A)/n(S)=18/36=1/2

P(B)=n(B)/n(S)=10/36=5/18

(b)

A and B ={(3,6),(4,5),(5,4)(5,6),(6,3),(6,5)}

n(A and B)=6

A or B= {{(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),

(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),

(5,2),(5,4),(5,6),(6,1),(6,3),(6,5),

(4,6)(5,5),(6,4),(6,6)}

n(A or B) =22

P(A and B)=n(A and B)/n(S)=6/36=1/6

P(A or B)=n(A or B)/n(S)=22/36=11/18

(c)Two events are INDEPENDENT if:

P(A AND B)=P(A)P(B)

P(A)=n(A)/n(S)=18/36=1/2

P(B)=n(B)/n(S)=10/36=5/18

P(A)P(B)=5/36

Since

P(A AND B)=1/6≠5/6=P(A)P(B)

The events A and B are not independent.

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