Respuesta :
Using the binomial distribution, there is a 0.9846 = 98.46% probability that at least one fund out of 4,170 funds outperforms the market in all 10 years, which means that it is highly likely that at least one fund out of 4,170 funds outperforms the market in all 10 years.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
Researching this problem on the internet, first we want to find the probability that one fund outperforms the market in all ten years, considering that on a given year they have a 50-50 chance, hence the parameters are:
p = 0.5, n = 10.
The probability is P(X = 10), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
P(X = 10) = (0.5)^10 = 0.001.
Then, we want the probability that at least one fund out of 4,170 funds outperforms the market in all 10 years, for which the parameters are given by:
n = 4170, p = 0.001.
The probability is:
P(X ≥ 1) = 1 - P(X = 0).
In which:
P(X = 0) = (1 - 0.001)^4170 = 0.0154.
Hence:
P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.0154 = 0.9846.
0.9846 = 98.46% probability that at least one fund out of 4,170 funds outperforms the market in all 10 years, which means that it is highly likely that at least one fund out of 4,170 funds outperforms the market in all 10 years.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
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