statistics

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 $n\to \mathrm{\infty }$ $$\sum _{i=1}^n{X}_i\sim N(n\cdot \mu ,n\cdot {\sigma }^2)$$ , 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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