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June 12th, 2011, 07:39 AM  #1 
Senior Member Joined: Dec 2010 Posts: 233 Thanks: 0  Probability
1.Three letters are selected at random from the word BIOLOGY.Find the number of ways in doing this.Hence,find the probability that the selection a)Does not contain letter O b)Contains both of the letter O Three letter select at random from the word BIOLOGY, i using 7C3 But the solution provided is 5C3 + 5C2 + 5C1 =25 I don understand why it is like that,and part (a) and (b) too. 2.Four person are chosen at random from a group of ten person consisting of four men and six women.Three of the women are sisters.Calculate the probability that the four person chosen will a)include the three sisters. This is the last part of the question,the other part i am able to solve.But having problem in this part.Thankk you. 
June 12th, 2011, 08:02 AM  #2  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Probability Hello, hoyy1kolko! Quote:
Four people are chosen from the available ten people. [color=beige]. . [/color] There are four men:[color=beige] .[/color] And six women, three of whom are sisters:[color=beige] .[/color] There is 1 way to choose the 3 sisters.  
June 12th, 2011, 08:40 AM  #3  
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Probability Quote:
Perhaps a more general approach would be to use generating functions. For each of the letters B,I,L,G,Y either you don't choose it or you do choose it. Represent that as For the O's you can either not choose any, choose 1, or choose 2. Represent this by Since you must do both tasks, multiply. and now you see every way to choose letters out of the seven. There are 25 ways to choose 3 letters, 16 ways to choose 5 letters, etc.  
June 12th, 2011, 02:39 PM  #4  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Probability Hello, hoyy1kolko! Quote:
mrtwhs gave a good explanation for their answer. But their answer is wrong! There are two O's and five Others. No O's: Choose 3 letters from the 5 Others. [color=beige]. . [/color] One O: Choose one O and 2 Others. [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] Two O's: Choose both O's and 1 Other. [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] Therefore, there are:[color=beige] .[/color] Quote:
We've already found the required quantities.  
June 12th, 2011, 03:12 PM  #5 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Probability
Actually their answer is RIGHT! 35 is the answer only if one of the O's is a capital O and the other O is lower case. Or maybe one of the O's is red and the other is blue. If, on the other hand, the two O's are indistinguishable then the correct answer is 25. 
June 12th, 2011, 03:54 PM  #6  
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Probability Quote:
If order is important, then you need to use an exponential generating function and the answer is 135 ways.  
June 12th, 2011, 04:33 PM  #7 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Probability
Here are the 25 ways assuming that order is not important. BIL,BIG,BIY,BLG,BLY,BGY,ILG,ILY,IGY,LGY BIO,BLO,BGO,BYO,ILO,IGO,IYO,LGO,LYO,GYO BOO,IOO,LOO,GOO,YOO If order is important then each pattern in the first two rows can be done in 6 ways (6 x 20 = 120) and the five in the last row can each be done in 3 ways (5 x 3 = 15) for a total of 135 ways. 
June 12th, 2011, 07:14 PM  #8  
Senior Member Joined: Apr 2011 From: USA Posts: 782 Thanks: 1  Re: Probability Quote:
Quote:
This is a common misconception in probabilities  it's a bit like the fact that a heads on one coin isn't the same as the heads on the other coin, even though they look alike. (And since the problem did not specify any type of order, I'm assuming order doesn't matter. But even if it did, it's still two different O's that could then have an order to them!)  
June 12th, 2011, 08:35 PM  #9  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Probability Hello, mrtwhs! Quote:
The objects do not have to be distinguishable. Suppose a bowl contains 999 white marbles and 1 black marble. We draw one marble at random. What is the probability that it is black? According to your reasoning, the 999 white balls are identical [color=beige]. . [/color]so there are two outcomes: (1) the ball is white, (2) the ball is black.  
June 13th, 2011, 02:41 AM  #10  
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Probability Quote:
You have four scrabble tiles N O O N. How many ways can you place them on your tray?  

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