If 2 out of 6 Machines are Defective, and 3 were selected randomly , What is the Probability Distrib

2nalfq8 2022-07-03 Answered
If 2 out of 6 Machines are Defective, and 3 were selected randomly , What is the Probability Distribution of the number of Defectives?
Even more generally :
If n objects are divided to m alike and k alike , we select x amount randomly , then to find how probable is it that y objects from x are m , using some trial and error I managed to find this complicated formula:
P X ( y ) = ( x y ) ( m P y ) ( k P x y ) n P x
But then it turned out that there is an easier formula with Combinations which is actually exactly equal to the previous one
P X ( y ) = ( m y ) ( k x y ) ( n x )
I can't understand how any of them can be shown mathematically or why do they have the same meaning.
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Answers (1)

Gustavo Brady
Answered 2022-07-04 Author has 9 answers
Explanation:
To help you understand the second formula consider this example.
Suppose a bag contains 8 Red and 4 Blue balls. 3 balls are chosen at random. Find the probability that 2 of the 3 are Red.
1. Total number of different choices of 3 from 12 = ( 12 3 ) = 220
2. Number of ways of choosing 2 Reds from 8 = ( 8 2 ) .
3. Number of ways of choosing 1 Blue from 4 = ( 4 1 )
4. Hence total number of ways of choosing 2 Reds and 1 Blue = ( 8 2 ) ( 4 1 ) = 112
5. Hence probability of choosing 2 Reds and 1 Blue = ( 8 2 ) ( 4 1 ) ( 12 3 ) = 112 220 .

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