# Probability that Alice and Bob keep dating infinitely often

I solved the following problem. I would appreciate it if you can please provide feedback and let me know if I have made any mistakes.

Problem statement:

Online dating: On a certain day, Alice decides that she will start looking for a potential life partner on an online dating portal. She decides that everyday, she will pick a guy uniformly at random from among the male members of the dating portal, and go out on a date with him. What Alice does not know, is that her neighbor Bob is interested in dating her. Being of a shy disposition, Bob decides that he will not ask Alice out himself. Instead, he decides that he will go out on a date with Alice only on the days that Alice happens to pick him from the dating portal, of which he is already a member. For the first two parts, assume that 50 new male members and 40 new female members join the dating portal everyday.

(a) What is the probability that Alice and Bob would have a date on the $$n^{th}$$ day? Do you think Bob and Alice would eventually stop meeting? Justify your answer, clearly stating any additional assumptions.

(b) Now suppose that Bob also picks a girl uniformly at random everyday, from among the female members of the portal, and that Alice behaves exactly as before. Assume also that Bob and Alice will meet on a given day if and only if they both happen to pick each other. In this case, do you think Bob and Alice would eventually stop meeting?

(c) For this part, suppose that Alice and Bob behave as in part (a), i.e., Alice picks a guy uniformly at random, but Bob is only interested in dating Alice. However, the number of male members in the portal increases by $$1\%$$ everyday. Do you think Bob and Alice would eventually stop meeting?

My attempt: My assumptions are the following.

1) Even if Alice has dated some men in the pool of available men to date in the past including Bob, the probability of Alice choosing Bob or any other man from the pool of men is still uniform. So even if Alice and Bob are neighbors and have dated on the second day for instance, the probability of Alice picking Bob on the fifth day is as if she absolutely no bias to meet Bob and she picks any man from the pool of men on the fifth day with a uniform probability. I assume the same for Bob for the second question.

2) The events that Alice choose Bob, Bob chooses Alice are independent. Also, the events where Alice chooses Bob on $$n^{th}$$ day and the event that Alice chooses Bob on $$k^{th}$$ day are independent if $$n \neq k$$.

a) On the $$n^{th}$$ day, there are $$50n$$ men. So if $$A_n$$ is the event that Alice chooses Bob, then $$P(A_n)=1/50n$$. Since $$\Sigma_{n=1}^{\infty}P(A_n) = \infty$$, they will date infinitely often from Borel-Cantelli lemma.

b) In this case, we need to define the event $$B_n$$ which is the set of outcomes where Bob and Alice both choose each other. Then $$P(B_n)=1/2000n^2$$. Since $$\Sigma_{n=1}^{\infty}P(B_n) = 1/2000 \times \pi^2/6 < \infty$$, they will not meet infinitely often from Borel-Cantelli lemma.

c) In this case, the set of outcomes $$C_n$$ is same as $$A_n$$ except that the pool of men has a number $$50 \times 1.01^n$$ on the $$n^{th}$$ day. Since $$\Sigma_{n=1}^{\infty}P(C_n) = 101/50 < \infty$$, they will not meet infinitely often from Borel-Cantelli lemma.

I just need to confirm if I have thought about this problem correctly and in case I have made mistakes, I would like to know about it. Please let me know.

• For me there are too many questions for one post. I recommend to split the questions into several posts. – callculus Mar 22 at 19:54
• @callculus I can do that but the meat of the problem will be common. So each question will still end up being too long. In case you have ideas to shorten the length of the problem, please feel free to edit and shorten it. – TryingHardToBecomeAGoodPrSlvr Mar 22 at 20:14
• I'm not sure, but I think that to answer the question you should also consider the probability of Alice and Bob never meet again after time $t$, and since that probability is zero in a) and in b), they will meet forever. Regarding c) it seems that the probability may not be zero, but I haven't computed the exact value. – Ertxiem Mar 22 at 23:04
• IMHO one post is fine: not only are the 3 questions are highly related, the OP is also using the same methodology i.e. Borel-Cantelli lemma, and the post is simply asking for verification that the OP is using the Borel-Cantelli lemma correctly in each case. It isn't asking for 3 different ways to solve 3 related questions. Anyway I am not an expert on Borel-Cantelli but it seems to me you're correct in applying it. I've upvoted this so someone who is actually an expert can answer definitively. – antkam Mar 25 at 14:09
• @antkam Thank you!! – TryingHardToBecomeAGoodPrSlvr Mar 25 at 18:36