An academic department with five faculty members narrowed...

An academic department with five faculty members narrowed its choice for department head to either candidate $A$ or candidate $B$ . Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for $A$ and two for $B .$ If the slips are selected for tallying in random order, what is the probability that $A$ remains ahead of $B$ throughout the vote count (e.g., this event occurs if the selected ordering is $A A B A B,$ but not for $A B B A A ) ?$

Key Concepts

Ballot Theorem

The Ballot Theorem is a key combinatorial result that determines the probability that one candidate is always ahead in vote counts when the total votes are known. It provides a formula that directly answers problems where the condition is to maintain a lead throughout a sequence of events, such as keeping one candidate in the lead during an election tabulation.

Combinatorial Enumeration

Combinatorial enumeration refers to counting and analyzing the number of possible arrangements that satisfy given constraints. In the context of vote sequences, it is used to calculate both the total number of ways votes can be ordered and the subset of those arrangements that adhere to the condition of a continuously leading candidate.

Sequential Probability Processes

Sequential probability processes involve studying events that occur in a specific order and determining the likelihood of certain outcomes throughout the sequence. This concept is essential in problems where the order of events, such as vote tallies, affects the overall probability of maintaining a lead.

Random Order Selection

Random order selection is the idea that each possible order of events is equally likely to occur when sampled without bias. This principle underpins the analysis of vote sequences since it allows the application of combinatorial and probability theories to determine the chance of one candidate staying ahead throughout the count.

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