Suppose there are a total of $N$ balloons in a closed room out of which n are blue and the rest are red. After every unit of time,say a minute, any two balloons collide with independent probability $p$.The rule is when a red and a blue balloon collide the blue one turns red; in all other cases nothing changes. What is the distribution of the number of red balloons after $t$ minutes? Here are my thoughts about the problem: If $m_i$ is the number of red balloons after $i$ minutes, then the probability $q_{i+1}$ that a blue balloon will have become red after $i+1$ minutes is given by $$1-\prod_{k=0}^{i}(1-p)^{(N-m_i)}.$$ Hence the probability that of there being $j$ red balloons after $i+1$ minutes is a binomial variate with parameters $q_{i+1}$ and $ N-m_i$. What I do not understand is how we can deal with probabilistically changing parameter $m_i$. Or, is there some better way of tackling the problem? Thanking you in anticipation!
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Do you have reason to expect that there should be a closed form for this? It doesn't seem like there should be. – joriki May 02 '20 at 10:37
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Could you give a more precise description of "any two balloons collide with independent probability"? Are there exactly $N/2$ pair-wise collisions with a random division of the balloons into pairs? What exactly is the meaning of $p$? – user May 02 '20 at 10:52
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@joriki yeah it seems there might be a clised solution.I am just curious for other possible approaches – AgnostMystic May 02 '20 at 12:12
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@sajjadveeri: I guess my question wasn't really clear. The question was not so much whether it seems to you that there might be a closed-form solution but whether you have reason beyond what's apparent from the question itself to think it should have one – for instance, that you found it as an exercise in a book (since exercises in books tend to have solutions). What's the source of the question? – joriki May 02 '20 at 12:25
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1@joriki i just posed this question to myself .i do not know whether it has or does not have a closed solution – AgnostMystic May 03 '20 at 10:20
1 Answers
Hint
It is assumed that the probability of collision be actually $p= \lambda \Delta t$, and that in $\Delta t$ the probability of having two or more collision be negligible.
Consider the balloons to be labelled. If a collision occurs we have $N(N-1)$ ways to choose an ordered couple $(x_1,x_2)$ of balloons to collide, and $2 m_k (N-m_k)$ ways to arrange an ordered couple (red-blue) or (blue-red).
Therefore the probability of having a transition $m_k \to m_k +1$ is $$ P(\Delta m_{k} ) = p{{2m_{k} \left( {N - m_{k} } \right)} \over {N\left( {N - 1} \right)}} $$
That's the base mechanism of the spreading of a disease ( an interesting exercise in these times).
Now you can cast that into a transition matrix, which will turn out into the path of a particle between two absorbing barriers at $m=0$ and $m=N$, with steps $\{0,1 \}$, having probability $1-P(m), \, P(m)$ and therefrom work out the Markov chain.
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