Describe HOW you would calculate the probability of dealing five cards in Poker and getting Full House (e.g., three cards with the same number or face, and two cards with a different same number or face). Explain your description in terms of the rules of probability and the definitions of permutations and combinations described in our reading. End your description by stating the correct probability.

We can calculate the probability by counting the total amount of favourable hands that are full house and dividing it by the total amount of hands (this is because each specific hand has the same probability).

There are 52 cards in poker and we deal 5, thus the total amount of hands is equal to the total amount of ways to pick 5 elements from a set of 52, in other words, the combinatorial number of 52 with 5, [tex]{52 \choose 5} = 2598960[/tex] .

To get a full house, you need to choose the number that appears thrice (you have 13 possibilities) and after that, from the 12 of number remaining, choose the one that appears twice (two possibilities). Then, you choose the 3 suits (from the 4 that are available) for the 3 cards with the same number ( [tex] {4 \choose 3} = 4 [/tex] possibilities), and you choose the 2 suits from the cards with the number that appear twice ([tex] {4 \choose 2} = 6 [/tex] possibilities). This gives us a total of 13*12*4*6 = 3774 favourable cases.

Therefore, the probability of obtaining a full house from a dealing of 5 cards is 3774/2598960 = 0.00145.