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The path to the $(3,2)$ is $3+2 \choose 3$ or $3+2 \choose 2$.

Total path is $4+4 \choose 4$

And the probability is : $ \frac{3+2 \choose 3}{4+4 \choose 4}$ =

$ \frac{5 \choose 3}{8 \choose 4}$ = $\frac{10}{70}$ = $0.14$

I am not sure if I solved this problem corectly.

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  • 1
    $\begingroup$ You're missing part of the problem statement: How does the ant move through the coordinate system? Does it always move by $+1$ in either the $x$ or $y$ directions? $\endgroup$
    – Brian Tung
    Sep 3 '15 at 16:53
  • $\begingroup$ Steps are 1 the the right or 1 up. $(1,0)$ or $(0,1)$ $\endgroup$
    – user258209
    Sep 3 '15 at 16:59
  • $\begingroup$ @SandroBrljafa: Is "the ant will find food at $(3,2)$" the same as "the ant will pass through $(3,2)$" in the question? (Is food always there?) $\endgroup$
    – mathlove
    Sep 3 '15 at 17:30
  • $\begingroup$ @mathlove the food is always there $\endgroup$
    – user258209
    Sep 3 '15 at 17:31
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It is not correct. Note that there are $\binom{3}{1}$ ways from $(3,2)$ to $(4,4)$.

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  • $\begingroup$ $5 \choose 2$ * $3 \choose 1$ divided by 70? $\endgroup$
    – user258209
    Sep 3 '15 at 17:43
  • $\begingroup$ @SandroBrljafa: Yes, exactly. $\endgroup$
    – mathlove
    Sep 3 '15 at 17:44
  • $\begingroup$ thanks, you've saved my life today, exactly that problem was on the test :) $\endgroup$
    – user258209
    Sep 4 '15 at 11:46
  • $\begingroup$ @SandroBrljafa: You are welcome! $\endgroup$
    – mathlove
    Sep 4 '15 at 11:48

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