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You own 50 DVDs consisting of 25 comedies, 15 dramas, and 10 thrillers.
You randomly pick 4 movies to watch during a long train ride.
What is the probability that you pick at least one DVD of each type of movie?
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First we calculate the number of all possible combinations of 50 DVDs taken 4 at a time.
It is the number of combinations, of course,
= = 230300.
This number is the cardinality of the space of events in this problem.
Next, we calculate the number of all possible favorable outcomes.
The favorable outcomes are 3 disjoint sets (2C, 1D, 1T), (1C, 2D, 1T), (1C, 1D, 2T),
where "C" means comedy CD; "D" means drama CD; "T" means thriller CD.
The number of elements in the set (2C, 1D, 1T) is = 300*15*10 = 45000.
The number of elements in the set (1C, 2D, 1T) is = 25*105*10 = 26250.
The number of elements in the set (1C, 1D, 2T) is = = 16875.
So, the total number of the favorable outcomes is 45000 + 26250 + 16875 = 88125.
Finally, the probability under the problem's question is
P = = = = 0.3827 (rounded). ANSWER
Solved and explained.
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Hello, this instruction about using the complement formula is IRRELEVANT to the posted problem.