Respuesta :
Using the binomial distribution, it is found that there is a 0.0328 = 3.28% probability that at least 2 of them choose the same quote.
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.
In this problem, we have that:
- There are 6 students, hence n = 6.
- There are 20 quotes, hence the probability of each being chosen is p = 1/20 = 0.05.
The probability of one quote being chosen at least two times is given by:
[tex]P(X \geq 2) = 1 - P(X < 2)[/tex]
In which:
P(X < 2) = P(X = 0) + P(X = 1).
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{6,0}.(0.05)^{0}.(0.95)^{6} = 0.7351[/tex]
[tex]P(X = 1) = C_{6,1}.(0.05)^{1}.(0.95)^{5} = 0.2321[/tex]
Then:
P(X < 2) = P(X = 0) + P(X = 1) = 0.7351 + 0.2321 = 0.9672.
[tex]P(X \geq 2) = 1 - P(X < 2) = 1 - 0.9672 = 0.0328[/tex]
0.0328 = 3.28% probability that at least 2 of them choose the same quote.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
