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Independence versus Dependence

Applications involving conditional probability and determining whether events are independent.

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All Those "Pure Luck" Game Shows

Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

Like watching game shows? Not the ones that require any type of skills or intelligence on the contestant's part, but the ones that are purely based on chance. Did you know that you can actually improve your chances of winning if a little bit of probability knowledge is applied?

Let’s Make a Deal

Consider the following example from a game show called Let's Make a Deal. Named after its original host Monty Hall, it's a famous probability puzzle known as the Monty Hall Problem.

Suppose you're a contestant on this show. You're given the choice of three doors: behind one door is a car and behind the others, goats. You pick a door, say No. 1, 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 switch your choice and pick door No. 2 instead?” Is it to your advantage to change your mind?

Credit: Trisha Shears
Source: http://commons.wikimedia.org/wiki/File:African_Pygmy_Goat_005.jpg
License: CC BY-NC 3.0

Many people (as a matter of fact, as high as 87% according to a survey that was once done) tend to believe that the probability of winning a car is simply \begin{align*}\frac{1}{2}\end{align*} whether or not you switch your choice (since there are 2 doors and just 1 car). However, if you look at the situation more closely, you may be surprised to find out that you have a \begin{align*}\frac{2}{3}\end{align*} chance of winning the car if you switch, and only a \begin{align*}\frac{1}{3}\end{align*} chance if you don't! Try figuring out why this happens.

You can find a detailed explanation here: http://www.youtube.com/watch?v=mhlc7peGlGg

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Another very similar probability puzzle is called Bertrand's Box Paradox. Check out the video below to see an adaption and explanation of this famous problem.

https://www.youtube.com/watch?v=EX4jZMGvubE

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Image Attributions

  1. [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0
  2. [2]^ Credit: Trisha Shears; Source: http://commons.wikimedia.org/wiki/File:African_Pygmy_Goat_005.jpg; License: CC BY-NC 3.0

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