In this chapter, we began practising with counting arguments. One of the ways we
ID: 203129 • Letter: I
Question
In this chapter, we began practising with counting arguments. One of the ways we will use counting arguments is in thinking about diffusive trajectories. Consider eight particles, four are black and four are white. Four particles can fit left of a permeable membrane and four can fit right of the membrane. Imagine that due to random motion of the particles every arrangement of the eight particles is equally likely. Some possible arrangements are: BBBB|WWWW, BBBW|BWWW, WBWB|WBWB; the membrane position is denoted by |.
(a) How many different arrangements are there?
(b) Calculate the probability of having all four black particles on the left of the permeable membrane. What is the probability of having one white particle and three black particles on the left of the membrane. Finally, calculate the probability that two white and two black particles are left of the membrane. Compare these three probabilities. Which arrangement is most likely?
(c) Imagine that in one time instant a random particle from the left-hand side exchanges places with a random particle on the right-hand side. Starting with three black particles and one white particle on the left ofthe membrane, compute the probability that after one time instant there are four black particles on the left. What is the probability that there are two black and two white particles on the left, after that same time instant? Which is the more likely scenario of the two?
Explanation / Answer
(a) How many different arrangements are there?
Ans. 5 arrangements are there, BBBB|WWWW, BWWW|WBBB, BBWW|WWBB, BBBW|WWWB, WWWW|BBBB
(b)
probability of having all four black particles on the left of the permeable membrane= 1/5 , probability of having one white particle and three black particles on the left of the membrane= 1/5, calculate the probability that two white and two black particles are left of the membrane= 1/5
All three arrangements are equally likely, as there is a random motion of particles.
(c) probability that after one time instant there are four black particles on the left= 1/5
probability that there are two black and two white particles on the left, after that same time instant= 1/5
the two scenario are equally likely, as there is a random motion of particles.
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