A bag contains 13 marbles; 3 blue, 2 green and 8 red. Without replacement find the following probabilities.
a) P(both have the same color)
b) P(different colors)
c) P(G2|R1)
d) P(G2 or R2)
e) P(no red marbles)
f) P(G2 and B2)
To find the probabilities in this scenario, we need to understand the concept of probability and how it is calculated.
Probability is the likelihood or chance of an event occurring. In this problem, we have a bag containing 13 marbles, of which there are 3 blue, 2 green, and 8 red marbles. Without replacement means that after selecting one marble from the bag, it is not put back before selecting the next marble.
a) P(both have the same color):
To find the probability that both marbles have the same color, we need to consider the cases where we select two blue marbles, two green marbles, or two red marbles.
P(both blue) = (3/13) * (2/12) = 6/156 = 1/26
P(both green) = (2/13) * (1/12) = 2/156 = 1/78
P(both red) = (8/13) * (7/12) = 56/156 = 7/26
Therefore, the probability that both marbles have the same color (either blue, green, or red) is:
P(both have the same color) = P(both blue) + P(both green) + P(both red) = 1/26 + 1/78 + 7/26 = 9/26 or approximately 0.346.
b) P(different colors):
To find the probability that the two marbles have different colors, we can subtract the probability of both marbles being the same color from 1.
P(different colors) = 1 - P(both have the same color) = 1 - 9/26 = 17/26 or approximately 0.654.
c) P(G2|R1):
This probability is asking for the probability of selecting a green marble as the second choice, given that the first marble selected was red.
P(G2|R1) = (2/13) * (8/12) = 16/156 = 2/39
d) P(G2 or R2):
This probability is asking for the probability of selecting a green marble as the second choice or a red marble as the second choice.
P(G2 or R2) = P(G2) + P(R2)
P(G2) = (2/13) * (11/12) = 22/156 = 11/78
P(R2) = (8/13) * (5/12) = 40/156 = 5/39
P(G2 or R2) = 11/78 + 5/39 = 24/78 = 8/26 or approximately 0.308.
e) P(no red marbles):
To find the probability of selecting no red marbles, we need to consider the cases where both marbles are either blue or green.
P(no red marbles) = P(both blue) + P(both green) = 1/26 + 1/78 = 4/78 = 2/39
f) P(G2 and B2):
This probability is asking for the probability of selecting a green marble as the second choice and a blue marble as the second choice.
P(G2 and B2) = (2/13) * (3/12) = 6/156 = 1/26
I hope this explanation helped you understand how to calculate the probabilities in this problem!