Question

Kyoko has a deck of 30 cards, numbered 1 through 30. She draws a card and then returns it to the deck. She does this 8
times.
What is the probability that she doesn't draw the same card more than once?

Answer 1

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:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

• n is the number of trials.
• p is the probability of a success in a single trial.
• x is the number of successes.

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by:

$C_{n,x} = \frac{n!}{x!(n-x)!}$  

• First, we find the probability that a single card is not drawn more than once.

• She draws 8 cards, thus, $n = 8$.
• Each card has a 1 in 30 probability of being drawn, thus $p = \frac{1}{30} = 0.0333$.

The probability of a card not being drawn more than once is:

$P(X \leq 1) = P(X = 0) + P(X = 1)$

In which

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{8,0}.(0.0333)^{0}.(0.9667)^{8} = 0.7627$

$P(X = 1) = C_{8,1}.(0.0333)^{1}.(0.9667)^{7} = 0.2102$

Then

$P(X \leq 1) = P(X = 0) + P(X = 1) = 0.7627 + 0.2102 = 0.9729$

We want this to be true for all 30 cards, thus, the probability that none are drawn more than once is:

$(0.9729)^{30} = 0.4386$

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