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Vanubis1
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Joined: February 21st, 2011, 7:41 am

Probability Question

March 18th, 2013, 10:54 am

Here is a game which was offered to my daughter: Fish and Fisher.I explain briefly the rules.You have a boat with 2 fishers (red and green), 5 empty squares, a square with 4 fishes (pink, blue, orange and yellow), 5 empty squares and the sea.At each turn, you roll a dice with the 6 colors, if the color is red or green, the boat moves by one square, if the color corresponds to a fish, this fish moves by one square.When the boat reaches one or several fishes, it caughts them, if the fish reaches the sea, it's saved.First Question: What are the probabilities that 0,1,2,3 or 4 fishes being caught?Extension: If the boat have caught fishes, the colors corresponding to these fishes will move the boat.What are the new probabilities?Second Extension: One player chooses the fishs and the other the fishers, if one or several fishes reaches the sea, its color will be used for another fish at the discretion of the fishes player.What is his optimal choice and what are the new probabilities?
Last edited by Vanubis1 on March 25th, 2013, 11:00 pm, edited 1 time in total.
 
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bluetrin
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Joined: September 9th, 2005, 6:41 am

Probability Question

March 19th, 2013, 10:03 am

I started to look at it with a binomial approach:If you consider 1 fish: +1 when the boat moves -1 when 1 fish moveIf you consider 16 steps, the fish has been caught if this reaches 6.You can use the mirroring property to say that the probability of touching 6 is equal to the probability of finishing above or equal to 6 ?But then when you consider more than 1 fish the probabilities are not independent.
Last edited by bluetrin on March 18th, 2013, 11:00 pm, edited 1 time in total.
 
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Vanubis1
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Joined: February 21st, 2011, 7:41 am

Probability Question

March 19th, 2013, 10:34 am

Interesting answer.You need the probability of moving (2/3 for +1 and 1/3 for -1).So, not sure the mirroring property could be applied.
 
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arentMuskets
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Joined: February 4th, 2013, 8:42 pm

Probability Question

April 2nd, 2013, 5:16 pm

I think the answer for part 1 is something like:no fish: 46.0%1 fish: 28.4%2 fish: 15.5%3 fish: 7.4%4 fish: 2.6%This is a quick simulation. I don't know how to solve this analytically.
 
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Vanubis1
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Probability Question

April 3rd, 2013, 8:54 am

Your answer is wrong.To check it quickly, as Bluetrin mention in a previous answer, it's quite easy to solve the problem for one fish and the probability to catch it is 59.77%.So in the 4 fishes problem, the average number of fish caught must be 4*59.77%=2.3906In your answer, it's 0.92PS: It's possible to solve the first part analytically.
Last edited by Vanubis1 on April 2nd, 2013, 10:00 pm, edited 1 time in total.
 
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arentMuskets
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Joined: February 4th, 2013, 8:42 pm

Probability Question

April 3rd, 2013, 1:00 pm

Yep, as usual I completely misread the problem. I read as there being two boats-- each propelled independently by the roll of their color. In reality there is only one boat, and either red or green moves it.Now that I've correctly read the problem, the answer is:No Fish: 8.8%1 Fish: 17.1%2 Fish: 23.6%3 Fish: 26.9%4 Fish: 23.5%And this does in fact imply that the average number of fish caught is 2.3906.Still probably worth it to solve it analytically, though.
 
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Vanubis1
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Probability Question

April 3rd, 2013, 1:32 pm

Right.Analytically, I have for the 1st question4 Fishes 23,50%3 Fishes 26,88%2 Fishes 23,65%1 Fish 17,12%0 Fish 8,85%and for the second4 Fishes 34,72%3 Fishes 24,71%2 Fishes 18,25%1 Fish 13,47%0 Fish 8,85%Last question is the more complex...
Last edited by Vanubis1 on April 2nd, 2013, 10:00 pm, edited 1 time in total.
 
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henktijms
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Joined: December 25th, 2010, 2:50 pm

Probability Question

April 8th, 2013, 9:33 am

Vanubis1, how did you analytically solve this problem? The only analytical solution method I see is using a recursion for a five-dimensional Markov chain.
 
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Vanubis1
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Probability Question

April 8th, 2013, 11:31 am

That's the way I've used.It's not too tricky because the number of nodes in 5 dimensions is not too high and it's the only way to answer to the last question.