Competition problem — Combinatorics

Library / /34 of 48

Combinatorics Difficulty 4.9 AIME Prove it United States

Problem:

Frank and Joe are playing ping pong. For each game, there is a $30\%$ chance that Frank wins and a $70\%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match?

Solution

Solution:

The expected value of the ratio of Frank's to Joe's score is $3:7$ , so Frank is expected to win $9$ games for each of Frank's $21$ . Thus the expected number of games in a match is $30$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.