b If two events A and B are dependent, then the probability that A and B will occur is P(AandB)=P(A)* P(B|A).
A
aProbability: 5/32 or ≈ 0.156
B
bProbability: 1/6 or ≈ 0.167
Comparison: It is 0.9 times less likely to draw red then green when replacing.
Practice makes perfect
a We randomly select marbles from a bag and we replace the first marble before drawing the second. In the bag we have 5 red, 8 green, and 3 blue marbles. Let's add these numbers to find the total number of marbles.
5 + 8 + 3 = 16
We are interested in finding the probability that we will randomly select a red marble first and then a green one. Let the event of selecting the red marble be A, and selecting the green marble be B. Therefore, we want to obtain the following probability.
P(A and B)
We need to decide whether events A and B are dependent or independent. Since we replace the first marble before we select the second one, the occurrence of the first event does not affect the occurrence of the second. Therefore, they are independent.
Probability of Independent Events
If two events A and B are independent, then the probability that A and B will occur is
P(AandB)=P(A)* P(B).
In order to find P(A) and P(B), we will use the theoretical probability. We need to compare the number of favorable outcomes to the number of possible outcomes. Let's start with P(A).
P=Favorable Outcomes/Possible Outcomes
We calculated that the bag contains 16 marbles. This is the number of possible outcomes. Out of these, 5 marbles are red. This is the number of favorable outcomes. We have enough information to calculate P(A).
Since we replace the first marble before drawing the second, the number of possible outcomes in the second drawing is also 16 marbles. To find P(B), we need to find the ratio of the number of green marbles, which is 8, to the total number of marbles.
b This time, when we are randomly selecting the marbles from the bag, we do not replace the first marble before drawing the second. As in Part A, the event of selecting the red marble will be A and selecting the green marble be B. We want to find the same probability.
P(A and B)
Since we do not replace the first marble before we select the second one, the occurrence of the first event affects the occurrence of the second. Therefore, they are dependent.
Probability of Dependent Events
If two events A and B are dependent, then the probability that A and B will occur is
P(AandB)=P(A)* P(B|A).
From Part A, we already know that the probability P(A) is 516. The difference this time is that we need to calculate P(B|A). The number of possible outcomes in the second drawing is different than in the first one, because we do not replace the first marble.
16-1=15
Because we selected a red marble in the first drawing, there are still 8 green marbles that could be chosen. We have enough information to calculate P(B|A).
Next, we want to compare the probabilities from Part A and Part B.
cc
With Replacing & Without Replacing [0.5em]
P(A and B)=5/32 & P(A and B)=1/6
In order to compare these probabilities, we will divide the P(AandB) from Part A by the one obtained in Part B.
