#Probability

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dire parcel
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Four points A, B, C, and D are connected by six line segments S1, S2, S3, S4, S5, and S6 as shown in the diagram. A coin is flipped for each segment k (where k = 1, 2, 3, 4, 5, 6). If it lands heads, segment Sk is painted blue; if it lands tails, segment Sk is painted red.

  1. Calculate the probability that you can move from A to B using only the blue segments.

  2. Calculate the probability that exactly three segments are painted blue.

  3. Calculate the probability that you can move between any two points using only the blue segments.

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harsh forgeBOT
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lone rapids
dire parcel
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I tried to calculate probability of (i)S1 is blue, (ii)S4, S5 are blue, (iii)S2, S4, S5 are blue, (iv)S2, S3 are blue, ... but I noticed there's an overlap and I'm stuck

dire parcel
lone rapids
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okay, I have something that may be a bit rough and not clever at all, but at the moment I have nothing else

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use conditional probability: for each edge Sk, I call the event Sk="the edge Sk is in blue", and I compute P(connected) = P(connected | S1) P(S1) + P(connected | not S1) P(not S1) = P(connected | S1 and S2) P(S1) P(S2) + P(connected | S1 and not S2) P(S1) P(not S2) …

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maybe you can find a clever choice of edges to condition with in order to limit the height of the tree

dire parcel
fathom finchBOT
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@dire parcel has given 1 rep to @lone rapids

lone rapids
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@dire parcel ok, maybe the clever way is the following: P(A and B connected) = P(smallest path has length 1) + P(smallest path has length 2) + P(smallest path has length 3)

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there are respectively 1 path of length 1 (S1), 2 paths of length 2 (S4,S5 and S3,S2) and 2 paths of length 3 (S4,S6,S2 and S3,S6,S5)

dire parcel
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I've been thinking of P(can move using S1)+P(can move using S4)+P(can move using S3)-overlap but your idea seems much easier, thanks! I'll try that.

dire parcel
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And q2 is 6C3(1/2)^3(1/2)^3=20/64=5/16

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For the Q3 I have no idea but I'm guessing using the diagram's symmetry might be a clue

lone rapids
dire parcel
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Oh I see, thanks for everything though!

lone rapids
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yes, the diagram symmetry is a clue, but I have not thought of it

lone rapids
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this is the underlying model, and you might find a correction if you're stuck

dire parcel
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Thanks for the info!

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Also I now came up with a mathematical solution for Q3, there are two ways where you can't move between any points, (1) equal to or more than 4 line segments are painted red
(2)Only three line segments which make the outline of a triangles(eg triangle ADB) are painted blue
Caluculate each of them and then 1-(1)'s probability-(2)'s probability.

obsidian briar
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  1. 1/2
    3)1/64
dire parcel
obsidian briar
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Yes

dire parcel
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Thanks for helping me but I guess question 1 would be more than 1/2, because if S1 is painted blue which is 1/2 we can move between those two points, but there are other ways such as S1 is red but S2, S3 are blue.

dire parcel
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If I'm not mistaken

dire parcel
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+close

fallen rainBOT
# dire parcel +close
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