If you draw five cards at random from a standard deck of 52 cards, what is the probability that there are at least 4 distinct characters (letters or numbers)

Question

Vocaloid · Answer

@smokeybrown would you mind assisting?

SmokeyBrown · Answer

Hey kingdomratchet, and welcome to QuestionCove!
Let's restate this problem in a way that will make it easier to visualize in terms of probability: to fulfill the conditions of the problem, we have to draw four cards, and none of them can be the same value as any of the others.
So, the first card we draw, of course, can't have any matches, since it is only a single card. In other words, there is a 52/52 chance that the first card is "unique".
When the second card is drawn, 51 cards remain. 3 of them are the same value as the first card drawn, and 48 of them are different. So there is a 48/51 chance that the second card is "unique".
The third card has a 3/50 chance of matching the first card, as well as a 3/50 chance of matching the second card, which means there is a 44/50 chance that the third card is unique.
Lastly, the fourth card has a 3/49 chance of matching the first card, a 3/49 chance of matching the second card, and a 3/49 chance of matching the third card, which leaves a 40/49 chance of being unique.
So, for each of these four conditions to be true together, the probability would be the product of the probabilities of the individual conditions.
That is, (52/52 * 48/51 * 44/50 * 40/49)
That should be the probability that all four cards have unique face values.