A stack of cards consists of seven red and four blue cards. A second stack of cards consists of eleven red cards. A stack is selected at random and three of its cards are drawn. If all of them are red, what is the probability the rst stack was selected

0.1733 = 17.33% probability the first stack was selected.

Step-by-step explanation:

To solve this question, it is needed to understand conditional probability, and the hypergeometric distribution.

Hypergeometric distribution:

The probability of x sucesses is given by the following formula:

In which:

x is the number of sucesses.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

is the number of different combinations of x objects from a set of n elements, given by the following formula.

Conditional probability:

We use the conditional probability formula to solve this question. It is

In which

P(B|A) is the probability of event B happening, given that A happened.

is the probability of both A and B happening.

P(A) is the probability of A happening.

Probability of all red cards for the first stack:

For this, we use the hypergeometric distribution, as the cards all chosen without replacement.

7 + 4 = 11 cards, which means that .

7 red, which means that

3 are chosen, which means that

We want all red, so we find P(X = 3).

Conditional probability:

Event A: All red

Event B: From the first stack.

Probability of all red cards:

0.2121 of 50%(first stack)

1 of 50%(second stack). So

Probability of all red cards and from the first stack:

0.21 of 0.5. So

What is the probability the first stack was selected?

0.1733 = 17.33% probability the first stack was selected.