How do you calculate the odds of catching a cold from a person who interacted with someone that has a chance of already having a cold?

My friends and I posed this question and I'm really curious how something like this would be calculated. I'm not sure if I phrased it correctly in the title, but increasingly complex examples I came up with would be:

If person 1 has a 1.5% chance of catching a cold each time they interact with person 2, who has a 5% chance of already having a cold, what are the odds of person 1 catching a cold each time they interact with person 2?

If two people have a 1.5% chance of catching a cold each time they interact with others, and person 1 interacted with 50 people before interacting with person 2, what are the odds that person 2 could catch a cold each time they interact with person 1?

If three people people all have a 1.5% chance of catching a cold each time they interact with others, person 1 interacted with 50 people, then person 2 interacted with person 1 10 times, what are the odds of person 3 catching a cold each time they interact with person 2?

If person 1 has a 5% chance of already having a cold and persons 2 and 3 both have a 1.5% chance of catching a cold each time they interact with others, what are the odds of person 3 catching a cold each time they interact with person 2 if person 2 previously interacted with person 1 10 times?

• Welcome to math.SE. What are your thoughts? Unless you show that you have expended some effort thinking about the problem yourself, your question is quite likely to be downvoted or closed. – rogerl Jan 4 '18 at 14:54
• I've spent some time thinking about it and figured out a bit, but am not sure if I got it right. All I can really figure out well is that a person has a 98.5% chance of not catching with each interaction. So their odds of catching one would be 1 - 0.985^n, with n being the number of interactions. Where I get lost is how to calculate person 1's odds of catching a cold from person 2 when only 5% of their interactions are when person 2 has a cold. Or if I try to add person 3 into the mix to see their chances of catching a cold from person 2 who caught theirs from person 1. – Randle J Jan 4 '18 at 15:43

If person 1 has a 1.5% chance of catching a cold each time they interact with person 2, who has a 5% chance of already having a cold, what are the odds of person 1 catching a cold each time they interact with person 2?

P(catch cold | $n$ meetings) = P(you catch cold on $n$ meetings given he has cold) x P(he has cold)

$$P(C|n) = (.05)(1-(1-.015)^n)$$

Next

If two people have a 1.5% chance of catching a cold each time they interact with others, and person 1 interacted with 50 people before interacting with person 2, what are the odds that person 2 could catch a cold each time they interact with person 1?

Let $C_1$ denote the event the first guy has a cold.

$$P(C_1)=1-.985^{50}$$ Then the event second guy has cold, $C_2$, after $m$ interactions has the probability $$P(C_2)=1-(1-P(C_1)^{m})$$ or $$P(C_2)=1-(1-(1-.985^{50}))^{m}$$