My Math Forum quarrel's problem

 February 10th, 2009, 12:12 AM #1 Newbie   Joined: Feb 2009 Posts: 4 Thanks: 0 quarrel's problem Chen, Li, Zhang, and Wang were all friends at school. Subsequently each of the 6 subpairs meet up; at each of the six meetings the pair involved quarrel with some fixed probability p , or became firm friends with probability 1 - p , Quarrels take place independently of each other. In future, if any of the four hears a rumour, then she tells it to her firm friends only. If Chen hears a rumour, what is the probability that: (a) Wang hears it? (b) Wang hears it if Chen and Li have quarrelled? (c) Wang hears it if Li and Zhang have quarrelled? (d) Wang hears it if she has quarrelled with Chen?
 February 22nd, 2009, 03:54 PM #2 Senior Member   Joined: Dec 2008 Posts: 251 Thanks: 0 Re: quarrel's problem I drew a probability tree for the first problem. Let $\sim$ be the relation of firm friendship and let Chen be $C$, Li be $L$, Zhang be $Z$, and Wang be $W$. Let $q\,=\,1\,-\,p$ be the probability of firm friendship between any two of them. $\begin{array}{ccccccccccccc} &&& & & & & & \cdot & & & & \\ &&& & & & & \swarrow & & \searrow & & & \\ &&& & & & C\not\sim W & & & &\underline{C\sim W} & & \\ &&& & & \swarrow & & \searrow & & & & & \\ &&& & W\not\sim Z & & & & W\sim Z & & & & \\ && & \swarrow & \downarrow & & & & \downarrow & \searrow & & & \\ &&W\not\sim L & & W\sim L & & & & Z\not\sim C & & \underline{Z\sim C} & & \\ && & \swarrow & \downarrow & & & & \downarrow & \searrow & & & \\ &&L\not\sim C & &\underline{L\sim C} & & & & Z\not\sim L & & Z\sim L & & \\ &\swarrow&\downarrow & & & & & & & & \downarrow & \searrow & \\ L\not\sim Z&&L\sim Z & & & & & & & & L\not\sim C & &\underline{L\sim C} \\ &&\downarrow & \searrow & & & & & & & & & \\ &&Z\not\sim C & &\underline{Z\sim C} & & & & & & & & \\ \end{array}$ Since all the probabilities are independent of each other and of those of preceding statements in the tree, each $\sim$ will hold with probability $q$ in relation to the preceding location of the tree and each $\not\sim$ will hold with probability $p$. This gives us $q\,+\,p(q(q\,+\,p\cdot q\cdot q)\,+\,p(q(q\,+\,p\cdot q\cdot q)))$ for the probability that Wang hears the rumor.

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