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Sagot :
The probability that the collector gets at least one limited edition card if he buys 3 packs is 0.23.
What is Binomial distribution?
A common discrete distribution is used in statistics, as opposed to a continuous distribution is called a Binomial distribution. It is given by the formula,
[tex]P(x) = ^nC_x p^xq^{(n-x)}[/tex]
Where,
x is the number of successes needed,
n is the number of trials or sample size,
p is the probability of a single success, and
q is the probability of a single failure.
Given that a box of trading cards contains 24-packs of cards in it. And Only two of those packs contain limited edition cards. Therefore, the probability of finding a limited edition card will be,
[tex]P = \dfrac2{24} = \dfrac{1}{12}[/tex]
The probability of not getting a limited edition card will be,
[tex]q = \dfrac{24-2}{24} = \dfrac{22}{24} = \dfrac{11}{12}[/tex]
Now, using the binomial distribution, the probability can be found.
A.) The probability that a collector will find both limited edition cards if he buys only 2 packs is
[tex]P(x) = ^nC_x p^xq^{(n-x)}\\\\P(x=2) = ^2C_2 \cdot(\dfrac1{12})^2 \cdot (\dfrac{11}{12})^{(0)}\\\\P(x = 2) = 0.0069 \approx 0.007[/tex]
B.) The probability that he gets at least one limited edition card if he buys 3 packs can be written as,
The probability of at least a limited edition card
= 1 - Probability of not getting any limited edition card
The probability of getting no special edition card will be,
[tex]P(x) = ^nC_x p^xq^{(n-x)}\\\\P(x=0) = ^3C_0 \cdot(\dfrac1{12})^0 \cdot (\dfrac{11}{12})^{(3)}\\\\P(x = 0) = 0.77[/tex]
Now,
The probability of at least a limited edition card
= 1 - Probability of not getting any limited edition card
The probability of at least a limited edition card = 1 - P(x=0) = 1-0.77 = 0.23
Hence, the probability that the collector gets at least one limited edition card if he buys 3 packs is 0.23.
Learn more about Binomial Distribution:
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