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Sagot :
Using the hypergeometric distribution, it is found that there is a 0.1515 = 15.15% probability that you pick at least 3 green balls.
The balls are chosen without replacement, hence the hypergeometric distribution is used.
What is the hypergeometric distribution formula?
The formula is:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- N is the size of the population.
- n is the size of the sample.
- k is the total number of desired outcomes.
In this problem:
- There are 4 + 5 + 3 = 12 balls, hence N = 12.
- 5 of the balls are green, hence k = 5.
- 4 balls will be picked, hence n = 4.
The probability that you pick at least 3 green balls is:
[tex]P(X \geq 3) = P(X = 3) + P(X = 4)[/tex]
Hence:
[tex]P(X = 3) = h(3,12,4,5) = \frac{C_{5,3}C_{7,1}}{C_{12,4}} = 0.1414[/tex]
[tex]P(X = 4) = h(4,12,4,5) = \frac{C_{5,4}C_{7,0}}{C_{12,4}} = 0.0101[/tex]
Then:
[tex]P(X \geq 3) = P(X = 3) + P(X = 4) = 0.1414 + 0.0101 = 0.1515[/tex]
0.1515 = 15.15% probability that you pick at least 3 green balls.
You can learn more about the hypergeometric distribution at https://brainly.com/question/4818951
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