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30 次抛硬币：正面数末位的期望

Coin is tossed 30 times

专题
Probability / 概率
难度
L4

题目详情

What is the expected last digit of the number of heads obtained when a fair coin is tossed 30 times?

解析

设 S∼Bin(30,12)S\sim\mathrm{Bin}(30,\tfrac12) 为正面数，所求为

E[S mod 10]=∑k=030(k mod 10)⋅(30k)230.\mathbb{E}[S\bmod 10]=\sum_{k=0}^{30}(k\bmod 10)\cdot \frac{\binom{30}{k}}{2^{30}}.

直接汇总同余类可得

E[S mod 10]=31676618567108864≈4.720184.\boxed{\mathbb{E}[S\bmod 10]=\frac{316766185}{67108864}\approx 4.720184}.