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Sumo Wrestlers
Part 1

Two Sumo wrestlers (A and B) have one wrestling match each day for NN days. Let XiX_i denote the outcome of a wrestling match on day ii for A (0 if A lost and 1 if A won) and let Xˉn\bar{X}_n denote the fraction of matches that A won.

Assume that each match is independent of the other matches and let pp denote the probability that sumo A wins the match. If p=0.63p = 0.63, and σ=0.025\sigma = 0.025 how many matches must the sumo wrestlers have in order for P(0.57Xˉn0.69)>0.88P(0.57 \leq \bar{X}_n \leq 0.69) \gt 0.88.