## Interview Question 4: Pigeon Hole

1 There are 10 people in a room. Each person shakes hand with 3 other people, what is the total number of handshakes? A Note: we don’t care if person A shakes with B, C, D or E, F, G. But every person reaches out three times. Thus, the total number of hand reach-outs is $$10 * 3 = 30$$ Every 2 reach-outs forms 1 handshake, thus total number of handshakes is 30 / 2 = 15.

## Interview Question 3: Conditional Probability

1 Q: Given 3 decks of cards, A to K. Randomly pick 1 card from each deck. What is the probability that the 3 cards are in increasing order? A $$\mathbb{P}(\text{3 different cards} \cap \text{3 increasing values}) = \mathbb{P}(\text{3 different cards}) \times \mathbb{P}(\text{3 increasing values} | \text{3 different cards})$$ First, make sure the three cards are different, $$\mathbb{P}(\text{3 different cards})$$ is $$1 * \frac{48}{52} * \frac{44}{52}$$.

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.