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My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get at least one call each day?

This question is 1.20 in Statistical interference by Casella Berger. The solution for which is given below by the author. The second sentence here makes me think the probability is 1. Where am I going wrong?

Solution

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closed as off-topic by Shaun, Shailesh, Claude Leibovici, user91500, N. F. Taussig Sep 9 '17 at 9:38

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    $\begingroup$ If you assume there is one call each day, then the answer is $1$. $\endgroup$ – Dave Sep 8 '17 at 20:02
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If each call is randomly assigned to a day, then this is equivalent to asking for the probability of finding each of $7$ coupons on $12$ draws in the Coupon collector's problem. It's also equivalent to asking the probability that a random $12$-letter word written with a $7$-letter alphabet contains all $7$ letters.

There are a total of $7^{12}$ words, and the number containing all $7$ letters is, by inclusion/exclusion:

$$7^{12}-\binom766^{12}+\binom755^{12}-\binom744^{12}+\binom733^{12}-\binom722^{12}+\binom711^{12}$$

This is an annoying calculation, but it can be done fairly simply on a spreadsheet, giving us a probability of:

$$\frac{3162075840}{7^{12}}\approx 22.846\%$$

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