My Math Forum Probability: Balls of same color

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 April 9th, 2013, 07:50 AM #1 Member   Joined: Apr 2013 Posts: 65 Thanks: 0 Probability: Balls of same color We have a bag that contains n different colored balls $(r_{1}, r_{2}, r_{3}, ... ,r_{n})$ The number of the balls are $a_{1}, a_{2}, a_{3}, ... ,a_{n}$ respectively. We pick n balls randomly (with replacement) $P(a_{1}, a_{2}, a_{3}, ... ,a_{n})$ is the probability of picking at least 2 balls with same colors. Which of the following is the least possible? $A) P(2,2,2,1) B) P(1,1,1,1) C) P(2,2,3) D) P(2,2,1) E) P(1,1,1)$ I don't know how to do this. If you could only show the calculation for the first option, I guess I can do the rest by myself.
 April 9th, 2013, 12:42 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Re: Probability: Balls of same color Can't "follow" your post (looks like I'm not the only one!): can you provide an example?
 April 9th, 2013, 12:59 PM #3 Member   Joined: Apr 2013 Posts: 65 Thanks: 0 Re: Probability: Balls of same color Alright. Let's say we have 4 different kind of balls: $r_{1}= red, r_{2} = blue, r_{3} = yellow, r_{4} = orange$ P(2,2,2,1) means we have 2 red, 2 blue, 2 yellow and 1 orange balls. We pick one ball at a time, put it back, do it 4 times. What is the probability of picking at least 2 same colored balls? An example: red,red,blue,yellow Is it $\frac{7^4-1}{7^4}$ or something?
 April 9th, 2013, 02:44 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 It seemed a bit strange that the number of different types of ball is also the number of balls that are picked, which may account for the lack of response. Also, you typed "least possible", but "least probable" would make more sense. This type of problem expects you to work out the value corresponding to each option by counting the number of ways of picking the n balls, and how many of those ways satisfy the specified criterion. It's not difficult to do this, so do give it a try. If you explain in detail what you can do (and why you did it), we can assess your work for you.

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