20 Consider the experiment called the birthday problem wher

20. Consider the experiment, called the birthday problem , where
our task is to determine the probability that in a group of people of a
certain size there are at least two people who have the same birthday
(the same month and day of month). Suppose there is a room with 6 people
in it, find the probability that at least two people have the same birthday.

24. A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Full house (one pair of the same face value and one triple of the same face value).

25. There are five Oklahoma State Officials: Governor (G), Lieutenant Governer (L), Secretary of State (S), Attorney General (A), and Treasurer (T). Take all possible samples without replacement of size 3 that can be obtained from the population of five officials. (Note, there are 10 possible samples!)

(a) What is the probability that the governor and the treasurer are included in the sample?
(b) What is the probability that the governor and the attorney general are included in the sample?

Solution

24)

Probability is the number of DESIRED cases out of the TOTAL number of cases.

The TOTAL number of cases is 52 C 5 or (52*51*50*49*48/5!)

The DESIRED number of cases is obtained by choosing a value for a pair, a value for the triple, and counting all possible suits.

We have 13 possible pairs.

After this, we have 12 possible triples because you cannot choose the same value as the pair.

For the pair, there are 4C2=6 possible face values.
For the triple. there are 4C1=4 possible face values.

So we have the DESIRED number of cases as 13*12*6*4.

The probability is thus 13*12*6*4/(52 C 5)= 0.144%