Statistics and Probability

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 13C5=1287. Some of these combinations consist of straight flushes. The 40straight flushes must be removed from the count. Thus there are 4 x13C5–40 = 5108 possible flushes.

Sincethere are 52 cards in the deck, then there are 52C51=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 4C2=6 in its rank and 4C3=4 choices for the 3OAK. Therefore there are 12 x 13 x 4C2 x 4C3=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., &amp Young, L. (2009). Introductionto Probability and its Applications.Cengage Learning.