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The sample space for this problem is the set $S=\{1,2,3,\ldots, 100\}$. The probability model is the equally likely outcome model. That is the probability that the number selected is in a subset $A\subseteq S$ is given by $$\mathbf{P}[A]=\frac{|A|}{|S|},$$ where $|\cdot |$ is number of elements in the set $\cdot$. If $A$ is a singleton, i.e. $|A|=1$, then $$\mathbf{P}[A]=\frac{1}{100}.$$ That is if you pick the correct number $0.01$. The alternative model you propose is that for any singleton $A$. $$\mathbf{P}[A]=\frac{1}{2}.$$ Let $A_i=\{i\}, i=1,2\ldots 100$. Clearly, $S=\cup_{i=1}^{100} A_i$ and $A_i\cap A_j=\emptyset$ if $i\not=j$. It follows from the finite additivity of the probability measure, that $$\mathbf{P}[S]=\mathbf{P}[\cup_{i=1}^{100}A_i]=\sum_{i=1}^{100}\mathbf{P}[A_i]=100\times \frac{1}{2}=50.$$ By the axioms of probability, for any event $B\subseteq S$, $0\leq \mathbf{P}[B]\leq 1$. Thus your proposed model violates the axioms of probability.
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Thu, 29 Jun 2023 17:53 GMT