I am working through some problems while studying and would like to validate my thinking with this problem.
There are six distinct deck of cards and one card is drawn from each deck. What is the probability that each card drawn is different?
I approached it as 1/52* 1/51 *… *1/47 because each time a card is drawn there is one less that is available to choose from if we don't want to match. I figured this is similar to 52 choose 1 times 51 choose 1 etc
Is this the correct way to approach it?
Best Answer
Your answer to this problem is not correct. In fact, the correct answer should be $$ \frac{52}{52} * \frac{51}{52} * \frac{50}{52} * \frac{49}{52} * \frac{48}{52} * \frac{47}{52} $$ This is how I got this answer - Consider event A to be the set of all outcomes in which all of the six cards are pairwise-distinct. Using combinatorics, this is equivalent to $$ 52*51*50*49*48*47 $$ The total number of ways in which 6 cards can be drawn from 6 decks is $$ 52^{6} $$ Thus by probability theory, the answer should be equal to the number of favourable outcomes divided by total outcomes. Hence $$ \mathbb{P}(A) = \frac{52}{52} * \frac{51}{52} * \frac{50}{52} * \frac{49}{52} * \frac{48}{52} * \frac{47}{52} $$ Hope it helps you!!!