Suppose 7 dice are rolled. What is the probability that at least 4 of the dice show the same face?
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.
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 be the number of successes when 7 dice are thrown. Then is . We have the following calculation:
Multiplying by 6 produces the desired answer.
To give some perspective to this example, for each , let be the event that at least 4 of the dice show the value of when 7 dice are rolled. The calculation above calculates the probability of the event . In this example, the event are mutually exclusive. This is the reason why we can multiply by 6 to obtained the answer in .
If we roll more dice, the event may no longer be mutually exclusive. For example, roll 8 dice and let be the event that at least 4 of the dice show the face of . Then the events are no longer mutually exclusive. To work this example, we need to use the multinomial theorem.
Answer to Example 2