You are lost on the planet of Skaro, where a species called Daleks live. Two thi

Question

You are lost on the planet of Skaro, where a species called Daleks live. Two thirds of the population on Skaro are inter-dimensional travellers, a species called Time Lords, who give a correct answer to a question with probability 3/4. Answers to repeated questions are independent even if the question and the individual are the same. If you ask a Dalek, the answer is always false.
[1] You ask a passer-by which species is superior, Daleks or Humans. The answer is Humans. What is the probability that this is correct?
[2] You ask the same passer-by again and receive the same reply. Show the probability that it is correct is 1/2.
[3] You ask the same passer-by again, and receive the same reply. What is the probability that it is correct?
[4] You ask for the fourth time, and received the answer Humans. Show that the probability it is correct is 27/70.
[5] Show that, had the fourth answer been Daleks instead, the probability that Humans is nevertheless correct is 9/10. You are lost on the planet of Skaro, where a species called Daleks live. Two thirds of the population on Skaro are inter-dimensional travellers, a species called Time Lords, who give a correct answer to a question with probability 3/4. Answers to repeated questions are independent even if the question and the individual are the same. If you ask a Dalek, the answer is always false.
[1] You ask a passer-by which species is superior, Daleks or Humans. The answer is Humans. What is the probability that this is correct?
[2] You ask the same passer-by again and receive the same reply. Show the probability that it is correct is 1/2.
[3] You ask the same passer-by again, and receive the same reply. What is the probability that it is correct?
[4] You ask for the fourth time, and received the answer Humans. Show that the probability it is correct is 27/70.
[5] Show that, had the fourth answer been Daleks instead, the probability that Humans is nevertheless correct is 9/10. You are lost on the planet of Skaro, where a species called Daleks live. Two thirds of the population on Skaro are inter-dimensional travellers, a species called Time Lords, who give a correct answer to a question with probability 3/4. Answers to repeated questions are independent even if the question and the individual are the same. If you ask a Dalek, the answer is always false.
[1] You ask a passer-by which species is superior, Daleks or Humans. The answer is Humans. What is the probability that this is correct?
[2] You ask the same passer-by again and receive the same reply. Show the probability that it is correct is 1/2.
[3] You ask the same passer-by again, and receive the same reply. What is the probability that it is correct?
[4] You ask for the fourth time, and received the answer Humans. Show that the probability it is correct is 27/70.
[5] Show that, had the fourth answer been Daleks instead, the probability that Humans is nevertheless correct is 9/10.

Explanation / Answer

1)P(answer is correct)=P(Time lord and answer is correct)+P(Dalek and answer is correct)

=(2/3)*(3/4)+(1/3)*0 =1/2

2)

P(same reply twice)=P(Time lord and correct answer both time)+P(Time lord and wrong answer both time)+P(Dalek and wrong answe both time)

=(2/3)*(3/4)*(3/4)+(2/3)*(1/4)*(1/4)+(1/3)*(1)*(1)=3/4

therefore P(correct|same reply twice)=P(Time lord and correct answer both time)/P(same reply twice)

=(2/3)*(3/4)*(3/4)/(3/4)=1/2

3)P(same reply 3 time)=P(Time lord and correct answer 3 time)+P(Time lord and wrong answer 3 time)+P(Dalek and wrong answe 3 time)

=(2/3)*(3/4)*(3/4)*(3/4)+(2/3)*(1/4)*(1/4)*(1/4)+(1/3)*(1)*(1)*(1)=5/8

P(correct|same reply thrice)=P(Time lord and correct answer thrice)/P(same reply thrice)

==(2/3)*(3/4)*(3/4)*(3/4)/(5/8)=9/10

4)

P(same reply 4 time)=P(Time lord and correct answer 4 time)+P(Time lord and wrong answer 4 time)+P(Dalek and wrong answe 4 time)

=(2/3)*(3/4)*(3/4)*(3/4)*(3/4)+(2/3)*(1/4)*(1/4)*(1/4)*(1/4)+(1/3)*(1)*(1)*(1)*(1)=35/64

P(correct|same reply 4 times)=P(Time lord and correct answer 4 time)/P(same reply 4 times)

=(2/3)*(3/4)*(3/4)*(3/4)*(3/4)/(35/64)=27/70

5)

P(4th answer is Dalek)=(2/3)*(3/4)*(3/4)*(3/4)*(1/4)+(2/3)*(1/4)*(1/4)*(1/4)*(3/4) =5/64

hence P(Human Correct|4th answer is Dalek)=(2/3)*(3/4)*(3/4)*(3/4)*(1/4)/(5/64)=9/10

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