Question

Part 1: In two or more complete sentences, explain how to find the probability of being dealt three kings from a standard deck of 52 cards when three cards are dealt. In your final answer, include all necessary steps and calculations.
Part 2: In two or more complete sentences, make a decision and support your position: Is Pascal’s triangle an appropriate method to determine the probability in Part 1?

Answer 1

1) We have that there are (52,3) ways to pick 3 cards out of 52 cards. Also, there are (4,3) ways to pick 3 kings out of the 4 total available kings in the deck.  In essence, we need to have one of those ways to be the selected 3 cards. Hence, the probability is the ration (4,3)/(52,3). Computing this:
P=$\frac{1*2*3*4}{(1*2*3)*1} / \frac{52*51*50}{1*2*3} = \\ \frac{4*1*2*3}{52*51*50}=0.0181%$
The probability is very low but not negligible.

2) The Pascal triangle defines a recursive relationship. Hence, we would need to calculate all the binomial coefficients up to 51. Thus, it is not at all practical to use the Pascal Triangle to calculate the ways. It is easier to do the direct computation.

Answer 2

Answer with explanation:

Part 1

Number of Kings in the deck of 52 card =4 kings

Number of ways of selecting 3 kings from 4 kings

                           $=_{3}^{4}\textrm{C}\\\\=\frac{4!}{(4-3)!\times 3!}\\\\=\frac{4 \times 3!}{3!}\\\\=4$    

Probability of an event

                        $=\frac{\text{Total favorable outcome}}{\text{Total Possible Outcome}}$

Probability of selecting three kings from a standard deck of 52 cards

                           $=\frac{_{3}^{4}\textrm{C}}{_{3}^{52}\textrm{C}}\\\\=\frac{4}{\frac{52!}{49!*3!}}\\\\=\frac{4}{22100}\\\\=\frac{1}{5525}$

Part 2:  

When you use the word Pascal's triangle , it is used to expand the expansion containing two or more terms having whole term raised to power n, where n is a positive Integer.No, Pascal's triangle is not an appropriate method to determine the probability in Part 1.It will be Lengthier.