Let's say we have a standard deck of 52 cards.
What would be the probability of choosing the 2 of diamonds?
Obviously, it would be $\frac{1}{52}$.
If we were to randomly choose another card from the deck, the probability of choosing the 2 of diamonds would be $\frac{2}{52}, or \frac{1}{26}$.
If we keep going on with this (while replacing each picked card), the probability of picking the 2 of diamonds will increase to $\frac{52}{52}$, or 100%.
Logically, we know that it is possible to pick 52 cards from a deck and put them back, and yet still not pick the 2 of diamonds.
So, practically speaking, what would be the probability of picking the 2 of diamonds at least once when picking a card from the deck 52 times, and how would I figure this out?
Best Answer
Same answer as Brian Fitzpatrick, but with IMHO a simpler explanation. The probability that you do not pick the $\diamondsuit2$ on the first draw is $\frac{51}{52}$. The same goes for all $52$ draws. These events are independent, so the probability that you never get the $\diamondsuit2$ in $52$ draws is $(\frac{51}{52})^{52}$, and the probability that you do get it at least once is $$1-\Bigl(\frac{51}{52}\Bigr)^{52}\ .$$