Respuesta :
Using the binomial distribution, it is found that there is a 0.4386 = 43.86% probability that she doesn't draw the same card more than once.
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Binomial probability distribution
It is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
- n is the number of trials.
- p is the probability of a success in a single trial.
- x is the number of successes.
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by:
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
- First, we find the probability that a single card is not drawn more than once.
- She draws 8 cards, thus, [tex]n = 8[/tex].
- Each card has a 1 in 30 probability of being drawn, thus [tex]p = \frac{1}{30} = 0.0333[/tex].
The probability of a card not being drawn more than once is:
[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8,0}.(0.0333)^{0}.(0.9667)^{8} = 0.7627[/tex]
[tex]P(X = 1) = C_{8,1}.(0.0333)^{1}.(0.9667)^{7} = 0.2102[/tex]
Then
[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.7627 + 0.2102 = 0.9729[/tex]
We want this to be true for all 30 cards, thus, the probability that none are drawn more than once is:
[tex](0.9729)^{30} = 0.4386[/tex]
0.4386 = 43.86% probability that she doesn't draw the same card more than once.
A similar problem is given at https://brainly.com/question/24756209
