i How many different ways can the podium be arranged? (The podium has a spot for only the 1st, 2nd and 3rd place horse) ii How many different ways can the horses finish the race such that Grand Valor, Seabiscuit and Secretariat do not finish first?
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.
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Notes to consider:
For part 1, A podium with (1st: horse A; 2nd: horse B; 3rd: horse C) is different than a podium with (1st: horse B; 2nd: horse C; 3rd: horse A). We're asking how many ways the 1st 2nd and 3rd horses could be. Let H = {h_0, h_1, ... h_11} be the 12 horses in the race. Here are two example ways the "podium" can be arranged: (h_3, h_7, h_2), (h_9, h_0, h_11)
For part 2,
Assume that horses may not share a placement (i.e. they can't "tie"). "Grand Valor, Seabiscuit and Secretariat do not finish first" means that: Grand Valor did not finish first AND Seabiscuirt did not finish first AND Secretariat did not finish first
For Part 3,
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