STATISTICS AND PROBABILITY9

Statisticsand Probability

Question1 a

Aflush contains all five cards from being of the same suit. Considerthe mathematical notation n Cr. Where _{}andr is the number of items that can be formed from a set of n distinctobjects(Scheafferet al. 2009). Introduction to Probability and its Applications.Cengage Learning..The possible number of five-card hands drawn fromeach of the fours suits is _{13}C_{5}=1287. Some of these combinations consist of straight flushes. The 40straight flushes must be removed from the count. Thus there are 4 x_{13}C_{5}–40 = 5108 possible flushes.

Sincethere are 52 cards in the deck, then there are _{52}C_{5}^{1}=2,598,960 possible combinations of five-card hands possible.

5108/ 2598960=0.0019654 thus gives probability of being dealt with aflush

Question1b

Afull house consists of the form of one pair and a three-of-a-kindwith 13 x 12 = 78 choices. There is a difference in which of the pairgets ranked. For instance, a full house constituting two 4s and three9s is different that consists of two 9s and three 4s. The choices forthe pair is _{4}C_{2}=6 in its rank and _{4}C_{3}=4 choices for the 3OAK. Therefore there are 12 x 13 x 4C2 x _{4}C_{3}=3744 possible full houses.

Sincethere are 52 cards in the deck, then there are 52C51= 2,598,960possible combinations of five-card hands possible.

Probabilityof being dealt with a full house is thus given by 3744 / 2598960=0.00144058

Question2 a

Inthis lottery drawing of a ball is random and without replacement. Theorder in which the balls are drawn does not matter. Thus, the numberof possible winning numbers is the combination of 6 objects selectedfrom 53, that is,

Thereis a need to ensure that each of the 22,957,480 possible winning areequally likely.The probability for a player who buys one ticket winsis

Question2 b

Thereare 23million people participating in the lottery, and only oneperson has to win from the 23million.

Probabilityof being a winner is given by,

Probabilityof not winning the Florida lottery is

_{}999

Question3a

Thefive white balls are drawn at random without replacement, and it isenough to match the numbers in any order. The possible ways ofchoosing 5 numbers from the 53 are the combinations of 5 objectsselected from 53:

Numberof ways of selecting a correct ball from six ball is:

Thenumber of possible winning numbers is from basic principle ofcounting, the product of the number of ways to select the five ballsand the number of ways to a correct ball:

Theprobability of winning the lottery is,

thatis less than the probability of winning Florida’s lottery.

Question3 b

Thenumber of ways to choose the 4 numbers from the 53 is thecombinations of 5 objects selected from 53:

Numberof ways of selecting a correct ball from six ball is:

Theprobability of winning the lottery is,

Thenumber of ways to choose the 3 numbers from the 53 is thecombinations of 5 objects selected from 53:

Numberof ways of selecting a correct ball from six ball is:

Theprobability of winning the lottery is,

Question3 c

Theprobability of winning some prize, by getting at least three of thesix winning numbers

=P (3 correct ) x P (4 correct ) x P (5 correct ) x P(6 correct

Question3 d

Theprobability that you win something,

Theprobability that you win nothing,

Question4a

Thenumber of possible winning numbers is the combination of 6 objectsselected from 53, that is,

Theprobability that a player who buys one ticket wins is

Theprobability that a player who buys 20 ticket wins is

Question4 b

Theprobability that a player is

Question4 c

Theprobability of winning is

Reference

Scheaffer,R., & Young, L. (2009). *Introductionto Probability and its Applications*.Cengage Learning.