Data Management Probability Distributions and Discrete Probability Question

Question

Hi, I am doing a question for my data management class, the unit is probabilty distrubtions and discrete probabilty and I would like to know how to go about it. A Euchre hand consists of 5 cards drawn without replacement from only 10s, Jacks, Queens, Kings and Aces. What is the expected number of spade(s) that will appear in a Euchre hand?

Answer 1

Rahman

The expected number of spades is the sum of (number of spades) x (probability of this number of spades).

The number of spades can be 0,1,2,3,4 or 5 so we need to find the probability of each of these.

Your 'pack' is 20 cards, 5 or which are spades and the other 15 are the other suits.

We can choose 5 cards from 20 in 20 C 5 ways = 20 ! / 5 ! 15 ! = 15504 ways.

How many have zero spades ? Well with zero spades we are choosing 5 out of the 15 non-spades. So number of ways is 15 C 5 = 15 ! / 5! 10 ! = 3003

1 spade. Well we can choose the spade in 5 C 1 ways, and the other 4 cards in 15 C 4 ways.

So number of ways = 5 C 1 x 15 C 4 = 5! / 1! 4! x 15 ! / 4 ! 11 ! = 6825 ways

2 spades, we can choose the spade in 5 C 2 ways, and the other 3 cards in 15 C 3 ways.

So number of ways = 5 C 2 x 15 C 3 ways = 10 x 455 = 4550 ways

3 spades . 5 C 3 x 15 C 2 = 10 x 105 = 1050

4 spades 5 C 4 x 15 C 1 = 5 x 15 = 75

5 spades 1

So the expectation will be 0x3003/15504 + 1x6825/15504 + 2x4550/15504 + 3x1050/15504 + 4x75/15504 + 5x1/15504 = 1/15504 x (0 + 6825 + 9100 + 3150 + 300 + 5) = 1.25

Mike