Alex the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves one unit up; if he flips tails, he moves one unit right. Betty the beetle starts at $(2,4)$. Each minute, she flips a fair coin. If she flips heads, she moves one unit down; if she flips tails, she moves one unit left. If the two start at the same time, what is the probability that they meet while walking on the grid?

  • 4
    $\begingroup$ Why is the ant just "Ant $A$" while Betty the beetle gets a name? The lack of consistency here is quite troubling. $\endgroup$ Apr 2, 2015 at 20:15
  • $\begingroup$ Um fine use Alex or something...does it really matter? $\endgroup$
    – user228320
    Apr 2, 2015 at 20:20
  • $\begingroup$ Have you tried making a list of the different possible combinations of moves that will result in the two insects meeting? There are not very many possibilities. $\endgroup$
    – David K
    Apr 2, 2015 at 20:21
  • $\begingroup$ Yes I have noted the 3 places where A and B could meet. But how would that help me find the probability? $\endgroup$
    – user228320
    Apr 2, 2015 at 20:24

1 Answer 1


They must meet at one of those points: $(0,3),(1,2),(2,1)$. each ant has a probability of $1/2^3$ to go through a specific path to one of the points, since they are 3-step from the start point.

Separate into complementary events:

meeting at $(0,3)$: "A" has 1 path, "B" has 3, we get: $P(0,3)=(1/8)*(3/8)$

meeting at $(1,2)$: Both has 3 paths: $P(1,2)=(3/8)*(3/8)$

meeting at $(2,1)$: same as the first

Summing the cases: $P=3/64 + 9/64 + 3/64 = 15/64$

  • $\begingroup$ isn't the probability of going through a particular path 1/7? $\endgroup$ Apr 2, 2015 at 20:36
  • $\begingroup$ @Tyler Hilton why? $\endgroup$
    – avim
    Apr 2, 2015 at 20:38
  • $\begingroup$ consider ant A. It has three ways to get to $(2, 1)$ and $(1, 2)$ but only 1 way to get to $(0, 3)$ which is a total of 7 ways. $\endgroup$ Apr 2, 2015 at 20:43
  • $\begingroup$ This is the number of ways it can walk to any point, not to a particular one. In order to walk in a particular path, say "up up right", it has to see heads,heads,tails, which has the probability of $0.5^3$ $\endgroup$
    – avim
    Apr 2, 2015 at 20:49

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