I've been working through some problems in Statistical Inference Second Edition (George Casella, Roger L. Berger), one of which is this already discussed problem. While the answer given makes sense, I'm having trouble understanding why the same probability can't be arrived at by calculating possible $\mathit{unordered}$ outcomes of calls mapped to days.
Question restated:
"My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get a least one call each day?"
While I am not interested in the answer, as it is already given, I am interested in why the following alternative approach doesn't work. What am I incorrect in assuming?
Number of possible ways the 12 indistinguishable calls could be arranged among 7 days:
$$\text{unordered with replacement} \Rightarrow \binom{7 + 12 – 1}{12} = \binom{18}{12}$$
To find the number of ways 12 calls could be spread among 7 days with at least one call per day, we first assign 1 call to each day. Next, the 5 remaining calls must be placed in some combination among the 7 days.
$$\text{again, unordered with replacement} \Rightarrow \binom{7 + 5 – 1}{5} = \binom{11}{5}$$
My thinking was then, having determined the total number of call-day combinations, as well as the total number of combinations with at least one call per day, the probability that each day had one call was trivially $\dots$
$$P(\text{one call per day}) = \frac{\binom{11}{5}}{\binom{18}{12}} = \frac{11}{442} \not\approx 0.2285$$
Where am I going wrong?
Best Answer
As a simpler example, consider the case of 3 calls and 2 days. Your method would calculate the probability of getting a call each day as $$\frac{|\{(1,2),(2,1)\}|} {|\{(0,3),(1,2),(2,1),(3,0)\}|} = \frac12.$$
This assigns equal probability to all ways of splitting three indistinguishable calls into two days. However, this is not the probability distribution the question implies. If each call is independently and uniformly distributed among the two days, the tuples $(0,3)$ and $(3,0)$ have probability $1/8$ each (since it means that all three calls went to the same day, so $(1/2)^3$), while $(1,2)$ and $(2,1)$ three probability $\binom31(1/2)^3 =3/8$ each.
To summarise, uniform distribution for each call is not the same as uniform distribution over compositions (of $12$ calls into $7$ parts).