Odd head count in a thousand tosses

Question

A fair coin is tossed 1000 times. What is the probability that the number of heads is odd?

Short answer

$1/2$ . Condition on the first 999 tosses: whatever the parity so far, exactly one outcome of the final toss makes the total odd, and it occurs with probability $\tfrac12$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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