(adsbygoogle = window.adsbygoogle || []).push({}); Conditional probability & "r balls randomly distributed in n cells"

1. The problem statement, all variables and given/known data

I'm posting this in hope that someone can give me a correct interpretation of the following problem (problem V.8 of Feller's Introduction to probability theory and its applications VOL I):

8. Seven balls are distributed randomly in seven cells. Given that two cells are empty, show that the (conditional) probability of a triple occupancy of some cells equals 1/4. Verify this numerically using table 1 of II,5.

2. Relevant equations

Conditional probability:

[tex]P\left \{A|B\right \} = \frac{P\left \{AB\right \}}{P\left \{B\right \}}[/tex]

Number of ways of distributing r indistinguishable balls in n cells:

[tex]\binom{n+r-1}{r}[/tex]

Number of ways of distributing r indistinguishable balls in n cells and no cell remaining empty:

[tex]\binom{r-1}{n-1}[/tex]

3. The attempt at a solution

I cannot arrive at the 1/4 figure, so I thought this may be due to a misunderstanding of the problem statement. The way I interpreted the problem is saying that the new sample space is the same as if any 2 of the seven cells turn out to be empty, giving 315 possible arrangements:

[tex]\binom{7}{2}\binom{7-1}{5-1} = 315[/tex]

With this scheme, only one cell of the remaining 5 can be triply occupied, because that leaves us with distributing the remaining 4 balls in the remaining 4 cells and no cell remaining empty. If we ignore the common factor 7 choose 2 = 21:

[tex]\frac{5}{\binom{7-1}{5-1}} = \frac{1}{3}[/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Conditional probability & r balls randomly distributed in n cells

**Physics Forums | Science Articles, Homework Help, Discussion**