For any tarot fans out there -
You have a 78 card deck. Each day you pick three cards. You repeat this for 200 days. What is the probability you pick the same pair of cards twice over the 200 days? How about three times?
To break it down further and add some math terms - each day you create 3 new pairs of cards (order does not matter - so let's say you draw cards A, B, and C, pairs AB, AC, and BC are now in your log to track if they appear again. You are not looking for any specific pair of two cards, just repeated ones). Population size 78, sample size 3, 200 trials, when you pick the three cards there is no replacement.
I can sort out the odds of picking a specific pair twice over the 200 trials, or three times over the 200 trials, but am having difficulty working through the probability of picking any pair twice or three times. Thank you!
To begin lets calculate the probability of choosing the same 3. The total combinations of 78 choose 3 = 76076. P(same 3 on day 2) = $\frac{1}{76076}$, P(same 3 on day 2) ∪ P(same 3 on day 3) = $\frac{2}{76076}$ ... P(same 3 in 200 days) = $\frac{199}{76076}$. Now considering pairs of cards, the total combinations of 79 choose 2 = 3003. P(same pair on day 2) = $\frac{1}{3003}$ * 3 as you have 3 chances to get pair. P(same pair on day 2) ∪ P(same pair on day 3) = ($\frac{1}{3003}$ *3) + ($\frac{1}{3003}$ * 3) = 6($\frac{1}{3003}$) = $\frac{6}{3003}$ ... P(same pair in 200 days) = 199 * $\frac{3}{3003}$ = $\frac{597}{3003}$ = $\frac{199}{1001}$. Generalised: P(same pair in n days) = (n-1)($\frac{3}{3003}$).