Question

Virus Z is a new virus infecting people. Some people get Virus Z and develop no systems. In others, Virus X causes a disease called Disease Z.

If you have Virus Z, you have it for life and everyone you touch gets Virus Z, if they didn’t have it already. (They get it immediately when you touch them.) If you get Virus Z, you will either immediately develop Disease Z, or you will never develop Disease Z, and those two outcomes are equally likely. Different people’s reactions to the virus are independent.

There is a 50% chance John has Virus Z. John does not have Disease Z. John touches four people without Virus Z. Given that the first three do not develop Disease Z, what is the probability that the fourth one develops Disease Z?

Answer 1

P(disease | virus) = 0.5

P(no disease | virus) = 0.5

P(john has virus) = 50% = 0.5

P(john doesn't have virus) = 1 - 50% = 0.5

P(three people don't develop disease | john has virus)

= P(no disease | virus)*P(no disease | virus)*P(no disease | virus)

= 0.5*0.5*0.5

= 0.125

P(three people don't have disease | john doesn't have virus) = 1

{if john doesn't have virus then 3 persons that he touches havve no chance of disease}

P(three persons no disease) = P(three people don't develop disease | john has virus)*P(john has virus) + P(three people don't have disease | john doesn't have virus) *P(john doesn't have virus)

= 0.125*0.5 + 1*0.5

= 0.5625

P(john has virus | three persons no disease) = P(three people don't develop disease | john has virus)*P(john has virus) / P(three persons no disease)

= 0.125*0.5 / 0.5625

= 0.11

P(4th develops disease | three persons no disease)

= P(develops disease | virus) * P(john has virus | three persons no disease)

= 0.50 * 0.11

= 0.055

P(4th develops disease | three persons no disease) = 0.055

(please UPVOTE)