a Calculate the relative frequency of event B occurring. To do so divide the total number of occurrences of event B by the total number of outcomes.
B
b Note that P(B|A) is a conditional probability, meaning that event B has occurred given that event A has occurred.
C
c Consider the probabilities obtained in the previous parts of the exercise. If they are the same, event A has no impact on the probability of event B.
D
d Calculate the probability of event D and the probability of event D, given that event C has occurred. Compare the probabilities to state if event C has any effect on the probability of event D.
E
e Calculate the probability of event F and the probability of event F, given that event C has occurred. Compare the probabilities to state if event C has any effect on the probability of event F.
A
a 7/16
B
b 7/16
C
c No. The events are independent.
D
d Yes, see solution.
E
e No, see solution.
Practice makes perfect
a Consider the given table.
B
D
F
Total
A
7
5
4
16
C
3
4
5
12
E
11
7
2
28
Total
21
16
11
48
First we want to calculate the probability of event B. To find this probability we will calculate the relative frequency. To do so, we need to divide the total number of occurrences of event B, 21, by the total number of outcomes, 48.
Relative Frequency=21/48=7/16
Therefore, the probability of event B occurring is 716.
b Now, we want to calculate the probability P(B|A). Note that this is a conditional probability, meaning that event B occurs given that event A has occurred. Consider the table once again.
B
D
F
Total
A
7
5
4
16
C
3
4
5
12
E
11
7
2
28
Total
21
16
11
48
The condition that event A has occurred limits the number of possible outcomes to 16. Events A and B occurred together 7 times. We can use these numbers to find the conditional probability.
P(B|A)=7/16
The probability of event B occurring given that event A has occurred is 716.
c Now we want to determine if the occurrence of event A has any effect on the probability of B. Consider the probabilities obtained in the previous parts of the exercise.
P(B)=& 7/16 [0.5em]
P(B|A)=& 7/16
The second probability means that event B occurred, given that event A has occurred. Note that these probabilities are the same, so event A does not have any effect on the probability of B. When the occurrence of one event has no effect on the other, the events are independent. Therefore, events A and B are independent.
d We will now state if events C and D are independent. To do so, we need to calculate the probability of event D, and the probability of event D occurring, given that event C has occurred. Consider the given table.
B
D
F
Total
A
7
5
4
16
C
3
4
5
12
E
11
7
2
28
Total
21
16
11
48
Let's start with calculating the probability of event D. The total number of occurrences of event D is 16, and the total number of possible outcomes is 48. To calculate the probability of event D we will divide these two numbers.
P(D)=16/48=1/3
We can now move on to the second probability, P(D|C). It is a conditional probability, meaning that event D occurs given that event C has occurred. The condition that event C occurs limits the number of possible outcomes to 12. Events C and D occurred together 4 times. We can use these numbers to find the conditional probability.
P(D|C)=4/12=1/3
Finally, we can compare the two obtained probabilities to see if event C has any impact on the probability of event D.
P(D)=& 1/3 [0.5em]
P(D|C)=& 1/3
Since the probabilities are the same, event C does not have any impact on the probability of D. Therefore, events C and D are independent.
e Finally, we will state if events C and F are independent. To do so, we need to calculate the probability of event F and the probability of event F occurring, given that event C has occurred. Consider the given table.
B
D
F
Total
A
7
5
4
16
C
3
4
5
12
E
11
7
2
28
Total
21
16
11
48
Let's start with calculating the probability of event F. The total number of occurrences of event F is 11, and the total number of possible outcomes is 48. To calculate the probability of event F we will divide these two numbers.
P(F)=11/48
We can now move on to the second probability, P(F|C). It is a conditional probability, meaning that event F occurs given that event C has occurred. The condition that event C occurs limits the number of possible outcomes to 12. Events C and F occurred together 5 times. We can use these numbers to find the conditional probability.
P(F|C)=5/12
Finally, we can compare the two obtained probabilities to see if event C has any impact on the probability of event F.
P(F)=& 11/48 [0.5em]
P(F|C)=& 5/12
Since the probabilities are different, event C has an impact on the probability of event F. Therefore, events C and F are not independent, so they are dependent.
