Answer:
0.4 = 40% probability that both marbles are blue.
0.0667 = 6.67% probability that both marbles are yellow.
53.33% probability of one blue and then one yellow
If you are told that both selected marbles are the same color, 0.8571 = 85.71% probability that both are blue
Step-by-step explanation:
To solve this question, we need to understand conditional probability(for the final question) and the hypergeometric distribution(for the first three, because the balls are chosen without being replaced).
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.
What is the probability that both marbles are blue?
4 + 2 = 6 total marbles, which means that
4 blue, which means that
Sample of 2, which means that
This is P(X = 2). So
0.4 = 40% probability that both marbles are blue
What is the probability that both marbles are yellow?
4 + 2 = 6 total marbles, which means that
2 yellow, which means that
Sample of 2, which means that
This is P(X = 2). So
0.0667 = 6.67% probability that both marbles are yellow.
What is the probability of one blue and then one yellow?
Total is 100%.
Can be:
Both blue(40%)
Both yellow(6.67%)
One blue and one yellow(x%). So
40 + 6.67 + x = 100
x = 100 - 46.67
x = 53.33
53.33% probability of one blue and then one yellow.
If you are told that both selected marbles are the same color, what is the probability that both are blue?
Conditional probability.
Event A: Both same color
Event B: Both blue
Probability of both being same color:
Both blue(40%)
Both yellow(6.67%)
This means that
Probability of both being the same color and blue:
40% probability that both are blue, which means that
Desired probability:
If you are told that both selected marbles are the same color, 0.8571 = 85.71% probability that both are blue