A popular gambling game called keno, first introduced in China over 2000 years ago, is played in many casinos. In keno, there are 80 balls numbered from 1 to 80. The casino randomly chooses 20 balls from the 80 balls. These are "lucky balls" because if a gambler chooses some of the numbers on these balls, there is a possibility of winning money. The amount that is won depends on the number of lucky numbers the gambler has selected. The number of ways in which a casino can choose 20 balls from 80 is C(80, 20) = 80! 20! · 60! ≈ 3,535,000,000,000,000,000. Once the casino chooses the 20 lucky balls, the remaining 60 balls are unlucky for the gambler. A gambler who chooses 5 numbers will have from 0 to 5 lucky numbers.Let's consider the case in which 2 of the 5 numbers chosen by the gambler are lucky numbers. Because 5 numbers were chosen, there must be 3 unlucky numbers among the 5 numbers. The number of ways of choosing 2 lucky numbers from 20 lucky numbers is C(20, 2). The number of ways of choosing 3 unlucky numbers from 60 unlucky numbers is C(60, 3). By the counting principle, there are C(20, 2) · C(60, 3) = 190 · 34,220 = 6,501,800 ways to choose 2 lucky and 3 unlucky numbers. Assume that a gambler playing keno has randomly chosen 4 numbers.In how many ways can the gambler choose exactly 2 lucky numbers?
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
A popular gambling game called keno, first introduced in China over 2000 years ago, is played in many casinos. In keno, there are 80 balls numbered from 1 to 80. The casino randomly chooses 20 balls from the 80 balls. These are "lucky balls" because if a gambler chooses some of the numbers on these balls, there is a possibility of winning money. The amount that is won depends on the number of lucky numbers the gambler has selected. The number of ways in which a casino can choose 20 balls from 80 is
| 80! |
| 20! · 60! |
Once the casino chooses the 20 lucky balls, the remaining 60 balls are unlucky for the gambler. A gambler who chooses 5 numbers will have from 0 to 5 lucky numbers.
Let's consider the case in which 2 of the 5 numbers chosen by the gambler are lucky numbers. Because 5 numbers were chosen, there must be 3 unlucky numbers among the 5 numbers. The number of ways of choosing 2 lucky numbers from 20 lucky numbers is
The number of ways of choosing 3 unlucky numbers from 60 unlucky numbers is
By the counting principle, there are
to choose 2 lucky and 3 unlucky numbers.
Assume that a gambler playing keno has randomly chosen 4 numbers.
In how many ways can the gambler choose exactly 2 lucky numbers?
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