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Seven balls are to be distributed randomly into seven cells. Let X4= # of cells containing exactly 4 balls.Determine the probability distribution of X4.

Is finding probability distribution is same as finding probability? If it is, my work; This can be done in the following ways; (4,3,0), (4,2,1), (4,1,1,1)

For example, the probability of happening (4,1,1,1)=

We can choose the cell that contains 4 balls in 7C1=7 ways. We can choose 4 distinguishable balls from 7 balls= 7C4 The 5th ball can be assigned to 6 different cells= 6C1 The 6th ball can be assigned to 5 different cells= 5C1 The 7th ball can be assigned to 4 different cells= 4C1 Also the last three balls can be distributed in 3! ways.

And our sample space is= 7.7.7.7.7.7.7= 7^7

So we multiply all these and repeat the process for the other ways. Is there anything wrong here?

Kamuran Karam
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1 Answers1

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To find the probability distribution of $X_4$ means to find the probability that $X_4=0$, the probability that $X_4=1$, the probability that $X_4=2$, and so on.

1) Since there are only $7$ balls, $P(X_4\ge 2)=0$.

2) $P(X_4=1)$ is the sum of the probabilities of three scenarios you mentioned. You are on the right track to calculate it.

3) $P(X_4=0) = 1- P(X_4=1)$, which you get from (2).

stochastic
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  • Then why sample space in the below link is n^n. https://math.stackexchange.com/questions/476114/probability-of-n-balls-in-n-cells-one-remaining-empty isn't that also supposed to be n! ? – Kamuran Karam Oct 21 '17 at 17:19
  • Nevermind, that was my mistake. I was thinking of identical balls. Fixed the answer. – stochastic Oct 21 '17 at 17:24
  • No really, I also think it has to be n!. I still don't get why it has to be n^n. Because once you put a ball in one of the cells regardless it is identical or not, it is gone, now you have n-1 balls. iSn't that so? What am I missing? – Kamuran Karam Oct 21 '17 at 17:25
  • If the balls are distinguishable, the first one can be placed in any of the 7 cells in 7 ways. Then the second ball can be placed in any of the 7 cells, and so on. – stochastic Oct 21 '17 at 17:28
  • First equation should be a strict inequality. Otherwise the second equation is also trivially equal zero, which would not be correct, as you can have a four-ball cell. – Nij Oct 21 '17 at 20:18