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Sagot :
Answer:
0.2143 = 21.43% probability that a contestant wins the game if he/she gets to select 4 of the buckets.
Step-by-step explanation:
The buckets are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
The probability of x sucesses is given by the following formula:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
8 covered buckets, so N = 8.
4 buckets are selected, so n = 4.
2 contain a ball, which means that k = 2.
Find the probability that a contestant wins the game if he/she gets to select 4 of the buckets.
This is P(X = 2). So
[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 = 2) = h(2,8,4,2) = \frac{C_{2,2}*C_{6,2}}{C_{8,2}} = 0.2143[/tex]
0.2143 = 21.43% probability that a contestant wins the game if he/she gets to select 4 of the buckets.
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