請教一題數學 有關 蒐集公仔數 期望值

蒐集第一個公仔只需拿 S1 = 1 個。得第一個公仔後,蒐集第二個公仔平均要再拿 : 1 次的機率是 3/4
2 次的機率是 (1/4)(3/4)
3 次的機率是 (1/4)²(3/4)
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n 次的機率是 (3/4) (1/4)^(n-1)故平均要再拿 S2 = 3/4 + 2(1/4)(3/4) + 3(1/4)²(3/4) + ...(1/4)S2 = (1/4)(3/4) + 2(1/4)²(3/4) + 3(1/4)³(3/4) + ...兩式相減 :S2 - (1/4)S2 = 3/4 + (1/4)(3/4) + (1/4)²(3/4) + (1/4)³(3/4) + ...(3/4)S2 = (3/4) / (1 - 1/4)S2 = 4/3蒐集得兩個公仔後,蒐集第三個公仔平均要再拿 : S3 = 2/4 + 2(2/4)(2/4) + 3(2/4)²(2/4) + 4(2/4)³(2/4) + ...S3 = 1/2 + 2(1/2)² + 3(1/2)³ + 4(1/2)^4 + ...(1/2)S3 = (1/2)² + 2(1/2)³ + 3(1/2)^4 + ...S3 - (1/2)S3 = 1/2 + (1/2)² + (1/2)³ + (1/2)^4 + ...(1/2)S3 = (1/2) / (1 - 1/2)S3 = 2 蒐集得三個公仔後,蒐集第四個公仔平均要再拿 : S4 = 1/4 + 2(3/4)(1/4) + 3(3/4)²(1/4) + 4(3/4)³(1/4) + ...(3/4)S4 = (3/4)(1/4) + 2(3/4)²(1/4) + 3(3/4)³(1/4) + ...S4 - (3/4)S4 = 1/4 + (3/4)(1/4) + (3/4)²(1/4) + (3/4)³(1/4) + ...(1/4)S4 = (1/4) / (1 - 3/4)S4 = 4 蒐集得全套四款公仔所需期望個數値 = S1 + S2 + S3 + S4= 1 + 4/3 + 2 + 4= 25/3

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冗凩儧剙傷

4~24吧?????????????????