Suppose that a six-sided die is \"loaded\" so that any particular even-numbered

Question

Suppose that a six-sided die is "loaded" so that any particular even-numbered face is twice as likely to be observed as any particular odd-numbered face.

(a) What are the probabilities of the six simple events? (Hint: Denote these events by O1,..., O6. Then P(O1) = p, P(O2) = 2p, P(O3) = p,..., P(O6)= 2p. Now use a condition on the sum of these probabilities to determine p. Answer as an exact fraction or round your answers to three decimal places.)
P(O1) =   
P(O2) =  
P(O3) =  
P(O4) =  
P(O5) =  
P(O6) =  

(b) What is the probability that the number showing is an odd number? (Give the answer to three decimal places.)


What is the probability that the number showing is at most 3? (Give the answer to three decimal places.)


(c) Now suppose that the die is loaded so that the probability of any particular simple event is proportional to the number showing on the corresponding upturned face; that is, P(O1) = c, P(O2) = 2c,..., P(O6) = 6c. What are the probabilities of the six simple events? (Answer as an exact fraction or round your answers to three decimal places.)
P(O1) =  
P(O2) =  
P(O3) =  
P(O4) =  
P(O5) =

  P(O6) =  

What is the probability that the number showing is an odd number? (Give the answer to three decimal places.)


What is the probability that the number showing is at most 3? (Give the answer to three decimal places.)

Explanation / Answer

a)as sum of probability is equal to 1

therefore p+2p+p+2p+p+2p=1

p=1/9

therefore P(O1)=1/9

P(O2)=2/9

P(O3)=1/9

P(O4)=2/9

P(O5)=1/9

P(O6)=2/9

b)

probability that the number showing is an odd number =(1/9+1/9+1/9)=3/9=1/3=0.333

probability that the number showing is at most 3=P(X<=3)=P(O1)+P(O2)+P(O3)=(1/9+2/9+1/9)=4/9=0.444

c)as sum of probability is equal to 1

c+2c+3c+4c+5c+6c=1

c=1/21

P(O1)=1/21

P(O2)=2/21

P(O3)=3/21

P(O4)=4/21

P(O5)=5/21

P(O6)=6/21

probability that the number showing is an odd number =(1/21+3/21+5/21)=9/21=0.429

probability that the number showing is at most 3=(1/21+2/21+3/21)=6/21 =0.286

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