Here is an example of a probability question illustrated with a deck of cards:

Sample Problem

How many ways are there to pick 5 cards from a deck with and without replacement (order matters)?


a) Ways to pick 5 cards with replacement

We are assuming in this question that a deck is standard and has 52 distinct cards. Since there is replacement, after each time you draw a card it is put back into the deck before you draw the next one.

Therefore, on each of the 5 draws you have 52 options.

So, the number of ways to draw 5 cards is 52 x 52 x 52 x 52 x 52 = 52^5.

b) Ways to pick 5 cards without replacement

Again the deck has 52 cards but this time if a card is drawn, it is not replaced before the next draw.

So for the first draw there are 52 options, but for the second draw there are only 51. For the third there are 50, the fourth 49 and the fifth 48.

So the number of ways to draw 5 cards is 52 x 51 x 50 x 49 x 48.

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