(3) Alice flips a fair coin n times and Bob flips another...

(3) Alice flips a fair coin $n$ times and Bob flips another fair coin $n+1$ times, resulting in independent $X \sim \operatorname{Bin}\left(n, \frac{1}{2}\right)$ and $Y \sim \operatorname{Bin}\left(n+1, \frac{1}{2}\right)$.
(a) Show that $P(X<Y)=P(n-X<n+1-Y)$.
(b) Compute $P(X<Y)$.

Key Concepts

Binomial Distribution

A binomial distribution describes the probability of having a fixed number of successes in a given number of independent trials, where each trial has the same probability of success. It is central to problems involving coin flips, dice rolls, and other binary outcomes, and its properties, such as mean, variance, and probability mass function, are crucial for calculating event probabilities.

Symmetry in Probability

Symmetry in probability involves the idea that certain events have equal likelihoods due to the inherent properties of the random process, such as the fairness of a coin. Recognizing symmetry allows one to equate seemingly different events or transformations, simplifying the process of calculating probabilities.

Change of Variables in Distributions

Changing variables in a probability distribution is a method used to transform a random variable into another form to simplify analysis or reveal hidden relationships. This technique can expose symmetry or equivalence in events by reinterpreting the original variable in a different, often more convenient, form.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways an event can occur relative to all possible outcomes. In the context of coin flips and binomial random variables, it is used to compute exact probabilities by enumerating the arrangements of successes and failures, which is a foundational tool in discrete probability theory.

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