A probability question.

Question

I had an improbable life event happen today and I want to figure out the chances of this occurring. I didn’t win the lottery in case you’re wondering

Let’s say there were 31,102 unique tokens in a bucket. Each day two people randomly choose one token. How many days would it take for these two peoples tokens to match up.

What would be the odds/probability of this event?

Additionally how would the answer change if the number was 23,145 instead.

The answer is really important to me, so thank you in advance!

Stephen Constantine

Answer 1

If any two token work:

The answer is $1/31102$. The first person picking out a marble doesn't influence the probability. Since it can be anything you want (all the marbles are desired). However, the second person has to get the same token which has a probability of $1/31102$. As there are $31102$ possibilities and only one is desired.

Similarly, $1/23,145$ is the probability.

If only a particular token works, the answer is as follows. $1/31102$ multiplied by $1/31102$. Since the first person has a chance of $1/31102$ and the second person has the same chance. Therefore we multiply them by each other. Think of it this way: If we want both people to select heads in a row with a coin. There are 4 possibilities: HH HT TH TT. Which corresponds to $1/2$ multiplied by $1/2$.

Answer 2

If all the tokens are unique, and they are replaced after each person chooses one, the probability of two consecutive people choosing the same token is $$\frac{1}{31,102}\times \frac{1}{31,102} = \frac{1}{967,334,404}~.$$ The same concept applies if the number of unique tokens were 23,145. The amount of days it would take, on average would be $967,334,404$.