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January 29th, 2017, 08:42 AM  #1 
Member Joined: Jan 2016 From: Blackpool Posts: 56 Thanks: 1  Homework question
Two balls are randomly chosen, without replacement, from an urn containing 8 blue, 4 yellow and 2 green balls. Suppose that we win £2 for each yellow ball selected and lose £1 for each blue ball selected. Let R be the random variable giving our profit. Find the induced sample space for R and evaluate the probabilities on the induced sample space. Hi guys please could you give me maybe a few pointers or hints on how to approach this question, I'm trying to work out the sample space first by using the 
January 29th, 2017, 09:34 AM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,600 Thanks: 816 
As all colors have at least 2 balls the possible selections can be $\{BB, YY, GG, BY, BG, YG\}$ The probability, for example, of $BB$ is $\dfrac{8}{14}\dfrac{7}{13} = \dfrac{4}{13}$ Similarly $P[YG] = \dfrac{4}{14}\dfrac{2}{13} = \dfrac{4}{91}$ I leave it to you to find the rest of these probabilities. Now as $R$ is the profit we get from the described scheme we can map each of these ball selection events into a profit event. It should be straightforward to see that the above set of pairs of ball colors maps to $R=\{2, 4, 0, 1, 1, 2\}$ with $P[2] = \dfrac {4}{13},~P[2]=\dfrac{4}{91}$, etc. So the set $R$ above is the induced sample space, and you'll finish finding each of their probabilities. 
January 30th, 2017, 03:33 AM  #3 
Banned Camp Joined: Nov 2016 From: St. Louis, Missouri Posts: 28 Thanks: 4 Math Focus: arithmetic, fractions 
In this scenario, one has a good chance of ending up with at least one blue ball and thus coming out ahead in terms of winning money.

January 30th, 2017, 12:19 PM  #4  
Senior Member Joined: May 2016 From: USA Posts: 823 Thanks: 335  Quote:
If you pick a blue ball, you lose money. Maybe you should change your slogan to "Todd gives math tips but does not guarantee that they are correct."  

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