Question

A box of of trading cards has 24-packs of cards in it. Only two of those packs contain limited edition cards.

Answer 1

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,

$P(x) = ^nC_x p^xq^{(n-x)}$

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,

$P = \dfrac2{24} = \dfrac{1}{12}$

The probability of not getting a limited edition card will be,

$q = \dfrac{24-2}{24} = \dfrac{22}{24} = \dfrac{11}{12}$

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

$P(x) = ^nC_x p^xq^{(n-x)}\\\\P(x=2) = ^2C_2 \cdot(\dfrac1{12})^2 \cdot (\dfrac{11}{12})^{(0)}\\\\P(x = 2) = 0.0069 \approx 0.007$

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,

$P(x) = ^nC_x p^xq^{(n-x)}\\\\P(x=0) = ^3C_0 \cdot(\dfrac1{12})^0 \cdot (\dfrac{11}{12})^{(3)}\\\\P(x = 0) = 0.77$

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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