Cartesian Skepticism

1 Q: Roll 100 dice together, what is the probability that the sum of all dice is 400? A Central Limit Theorem: Let \(X_1, X_2,... X_n\) be independent and identically distributed random variables. The sum of these random variables approaches a normal distribution as \(n \rightarrow \infty\) $$ \sum_{i=1}^n X_i \sim N(n \cdot \mu, n \cdot \sigma^2) $$ , where \(\mu = E[X_i]\) and \(\sigma^2 = Var(X_i)\). Let \(X\) be the value of a die.

1 Given a deck of 52 cards, only consider the color black and red. Shuffle the cards. Define that a group is a sequence of same-color cards. For example, Red/Black/Black/Black/Red/Red is an array of 3 groups. Q: What is the expected number of groups? A: Consider a simple case of 2 cards. If they have the same color, the \(E\) will be 1. \(E\) will be 2 if the 2 cards have different colors.

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