During a game of poker, you are dealt a five-card hand at random from a card deck. A card deck contains 52 cards. There are four suits, Spades, Clubs, Diamonds and Hearts. Each suit has 13 ranks.
Compute following probabilities:
You look over the shoulder of the person next to you for a second and find out she does not have any card of Spades. How can you update your estimate that she might have a pair in her hand? How about two pairs? Provide two methods of estimation. One by counting, similar to earlier parts, and other by using the Bayes Rule(20 points): \begin{equation} P(A|B) = \frac{P(B | A)\, P(A)}{P(B)} \end{equation}
A similar probabilistic scheme is widely used to play online Poker games. One of the most famous players using this method was Nate Silver, author of Signal and the noise. He gave up his job at a good paying financial institution to play online Poker games. You can read his story as well as many other stories in using probabilistic models for predicting future in this book.
Let X have distribution function
\begin{align} F_X(x) = \left\{ \begin{array}{l l} 0 & \text{if $x<0$,}\\ \frac{1}{2}x & \text{if $0 \leq x \leq 2$}\\ 1 & \text{if $x > 2$}\\ \end{array} \right. \end{align}Let $Y=X^2$.
Compute the following:
$\mathbb{P}(\frac{1}{2} < X \leq \frac{3}{2})$ (10 points)
$\mathbb{P}(Y < X)$ (10 points)
distribution function of $Z = \sqrt{X}$ (10 points)
What is the expected number of times one needs to throw a die to get a 6? (10 points)
Hint: There are two possible ways to solve this problem, one is by characterizing the number of times needed to get a six as a Geometric distribution and one by using law of total expectation.
This is a job interview question! of course without the hint. :-)