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Sagot :
Using the hypergeometric distribution, it is found that there is a 0.4286 = 42.86% probability of getting 2 of the same colour.
The marbles are chosen without replacement, hence the hypergeometric distribution is used to solve this question.
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 is a total of 3 + 4 = 7 marbles, hence N = 7.
- Of those, 3 are blue, hence k = 3.
- 2 marbles will be taken, hence n = 2.
The probability of getting 2 of the same colour is the sum of P(X = 0), which is both red, with P(X = 2), which is both blue, then:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 0) = h(0,7,2,3) = \frac{C_{3,0}C_{4,2}}{C_{7,2}} = 0.2857[/tex]
[tex]P(X = 2) = h(2,7,2,3) = \frac{C_{3,2}C_{4,0}}{C_{7,2}} = 0.1429[/tex]
Hence:
[tex]p = P(X = 0) + P(X = 2) = 0.2857 + 0.1429 = 0.4286[/tex]
0.4286 = 42.86% probability of getting 2 of the same colour.
You can learn more about the hypergeometric distribution at brainly.com/question/4818951
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