## A Binomial Example

Example 1
Suppose 7 dice are rolled. What is the probability that at least 4 of the dice show the same face?

Example 2
Suppose that 6 job assignments are randomly assigned to 5 workers. What is the probability that at least 4 of the job assignments go to the same worker?

Example 2 is left as exercise.

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Discussion of Example 1
Fix a face (say 1). Finding the probability of that at least 4 of the dice show the face 1 is a binomial problem. Then multiplying this answer by 6 will give the desired answer.

Consider obtaining a 1 as a success. Let $X$ be the number of successes when 7 dice are thrown. Then $X$ is $\text{binom}(7,\frac{1}{6})$. We have the following calculation:

\displaystyle \begin{aligned}(1) \ \ \ \ \ P(X \ge 4)&=1-P(X \le 3) \\&=1-P(X=0)-P(X=1) \\&- \ \ \ P(X=2)-P(X=3) \\&=1-\binom{7}{0} \biggl[\frac{1}{6} \biggr]^0 \biggr[\frac{5}{6} \biggr]^7 - \binom{7}{1} \biggl[\frac{1}{6} \biggr]^1 \biggr[\frac{5}{6} \biggr]^6 \\&- \ \ \ \binom{7}{2} \biggl[\frac{1}{6} \biggr]^2 \biggr[\frac{5}{6} \biggr]^5 - \binom{7}{3} \biggl[\frac{1}{6} \biggr]^3 \biggr[\frac{5}{6} \biggr]^4 \\&=\frac{4936}{279936} \end{aligned}

Multiplying $(1)$ by 6 produces the desired answer.

\displaystyle \begin{aligned}(2) \ \ \ \ \ 6 \times P(X \ge 4)&=6 \times \frac{4936}{279936} \\&=\frac{29616}{279936} \\&=0.105796 \end{aligned}

To give some perspective to this example, for each $i=1,2,3,4,5,6$, let $A_i$ be the event that at least 4 of the dice show the value of $i$ when 7 dice are rolled. The calculation $(1)$ above calculates the probability of the event $A_i$. In this example, the event $A_i$ are mutually exclusive. This is the reason why we can multiply by 6 to obtained the answer in $(2)$.

If we roll more dice, the event $A_i$ may no longer be mutually exclusive. For example, roll 8 dice and let $A_i$ be the event that at least 4 of the dice show the face of $i$. Then the events $A_i$ are no longer mutually exclusive. To work this example, we need to use the multinomial theorem.

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$\displaystyle \frac{1325}{15625}=0.0848$