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Betting with Coin Flips

Alice and Bob flip a coin repeatedly. Each time there is a head bob gets a dollar and each time there is a tail Alice gets a dollar.

  1. What is the probability that Bob and Alice have exactly the same amount of money after \(2n\) flips ?
  2. What is the chance that Alice has more money after \(2n+1\) flips ?

Pathway enrichment

A list of \(100\) genes are known to be part of the oxidative phosphorylation pathway.

My friend a molecular biologist screened the activity of \(5000\) genes in both diabetics and normal individuals.

He/she found \(500\) genes that were more active in normal individuals than diabetics.

Of these genes \(60\) of them belong to the list of genes that are part of the oxidative phosphorylation pathway.

What is the probability of this even happening randomly ? What is the scientific question behind the probability problem ?

Seating people

In how many ways can \(6\) people be seated in \(11\) vacant chairs that are arranged in a row ?

Inclusion of origin

Draw \(n\) points from the uniform distribution on the circle and draw the convex hull around these points. What is the probability that the origin (center of the circle) is contained in the convex hull ?

 

[From: The Probabilistic Method by Alon and Spencer]

Monty Hall

Consider the Monty Hall problem:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

(You have observed that week after week on this show, the host always opens a door to revival a goat before giving you the option to switch)

  1.  Without using conditional probabilities work out the probability of winning if you switch versus if you don’t.
  2.  Compute the same probabilities using conditional probabilities.
  3. Now assume that the host is cruel. He only offers you a switch when you have chosen the car. (You figure this out by watching the show on TV before going.) Now should you switch when offered the chance ?
  4. Now lets assume that every morning the host gets up and with probability \(p\) acts like the original host and with probability \(1-p\) he acts like the cruel host from the previous question.
    1. Now what is the chance  of wining if you switch ?
    2. Now what is the chance  of wining if you don’t switch ?
    3. For which \(p\) should you switch and which \(p\) should you not switch ?
  5. Now let us assume that host just picks a door randomly after you have picked yours. If there is a car behind it the came is over. If there is a goat, he offers you the chance to switch. Should you switch when you have the chance ? why ?

 

Counting and geometry

1) How many ways can one order \(n\) people on a line ?

2) How many ways can one order \(n\) people on a circle or round table (note that rotations are considered equivalent )?

3)  How many ways can one order \(n\) people on a circle or round table with invariance with respect to rotations and direction (clockwise versus counter-clockwise) ?

Statespace of the NCAA Basketball tourniment (in the old days)

  1. Until 2000,  the NCAA men’s Basketball tournament had 64 teams. They played each other  in 6 rounds With the winner of each game moving one to play the winner of another game in a pre-specified order. If we assume the initial table of “who plays who” as well as who each winner plays is specified, give a concise  description of the state space of all possible out comes. How many elements does the space have ?
  2. (*) If there are now \(2^n\) competitors and \(n\) rounds, answer the same questions as before.

 

[ Inspiration [GS2] p 1, # 3]

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