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Answer:
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.