From the sample space S={1,2,3,4,5,6,7,8}, a single number is selected randomly. Given the events
A: selected number is even
B: selected number is a multiple of 3
Find each probability.
P(A)=
P(B)=
P(A and B)=
P(B ┤| A)=
P(A ┤| B)=
Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if
P(B ┤| A)=P(B), or equivalently P(A ┤| B)=P(A).
Are the events A and B independent? Explain why or why not.
To find each probability, we need to count how many numbers satisfy the condition and divide by the total number of elements in the sample space.
1) P(A): The selected number is even.
There are 4 even numbers in the sample space: 2, 4, 6, and 8.
P(A) = Number of favorable outcomes / Total number of outcomes = 4 / 8 = 1/2
2) P(B): The selected number is a multiple of 3.
There are 2 numbers in the sample space that are multiples of 3: 3 and 6.
P(B) = Number of favorable outcomes / Total number of outcomes = 2 / 8 = 1/4
3) P(A and B): The selected number is both even and a multiple of 3.
There is only 1 number in the sample space that satisfies this condition: 6.
P(A and B) = Number of favorable outcomes / Total number of outcomes = 1 / 8 = 1/8
4) P(B | A): The probability of B given A.
P(B | A) = P(A and B) / P(A)
P(A and B) = 1/8 (as calculated in the previous step)
P(A) = 1/2 (as calculated in step 1)
P(B | A) = (1/8) / (1/2) = 1/4 / 1/2 = 1/4 * 2/1 = 1/2
5) P(A | B): The probability of A given B.
P(A | B) = P(A and B) / P(B)
P(A and B) = 1/8 (as calculated in step 3)
P(B) = 1/4 (as calculated in step 2)
P(A | B) = (1/8) / (1/4) = 1/8 / 1/4 = 1/8 * 4/1 = 1/2
A and B are independent events if P(B | A) = P(B) or P(A | B) = P(A).
In this case, P(B | A) = 1/2 and P(B) = 1/4, so they are not equal.
Similarly, P(A | B) = 1/2 and P(A) = 1/2, so they are equal.
Since P(B | A) is not equal to P(B), events A and B are not independent.
To find the probabilities for each event, we need to count the number of outcomes that satisfy the conditions and divide it by the total number of outcomes in the sample space.
1. P(A): The selected number is even.
Outcomes that satisfy the condition: 2, 4, 6, 8
Number of outcomes = 4
Total number of outcomes in the sample space = 8
P(A) = Number of outcomes that satisfy the condition / Total number of outcomes = 4 / 8 = 1/2
2. P(B): The selected number is a multiple of 3.
Outcomes that satisfy the condition: 3, 6
Number of outcomes = 2
Total number of outcomes in the sample space = 8
P(B) = Number of outcomes that satisfy the condition / Total number of outcomes = 2 / 8 = 1/4
3. P(A and B): The selected number is both even and a multiple of 3.
Outcomes that satisfy the condition: 6
Number of outcomes = 1
Total number of outcomes in the sample space = 8
P(A and B) = Number of outcomes that satisfy the condition / Total number of outcomes = 1 / 8 = 1/8
4. P(B ┤| A): The probability of B given that A has occurred. This is asking for the probability of B occurring, assuming that A has already occurred. Since all even numbers are divisible by 2, if a number is even, it is automatically a multiple of 2. Therefore, P(B ┤| A) = P(B).
5. P(A ┤| B): The probability of A given that B has occurred. This is asking for the probability of A occurring, assuming that B has already occurred. Since 6 is the only number that is both even and a multiple of 3, if a number is a multiple of 3, it is automatically even. Therefore, P(A ┤| B) = P(A).
A and B are considered independent events if P(B ┤| A) = P(B) or P(A ┤| B) = P(A).
In this case, since P(B ┤| A) = P(B) and P(A ┤| B) = P(A), the events A and B are independent. This means that the knowledge of one event does not affect the probability of the other event.
To find the probability of each event, we need to count the favorable outcomes and divide it by the total number of outcomes.
1. P(A) - Probability that the selected number is even:
Count the number of even numbers in the sample space, which are 2, 4, 6, and 8. So, the favorable outcomes for event A are 4.
Total outcomes in the sample space = 8 (since there are 8 numbers).
P(A) = favorable outcomes / total outcomes = 4/8 = 1/2 = 0.5
2. P(B) - Probability that the selected number is a multiple of 3:
Count the number of multiples of 3 in the sample space, which are 3 and 6. So, the favorable outcomes for event B are 2.
Total outcomes in the sample space = 8 (since there are 8 numbers).
P(B) = favorable outcomes / total outcomes = 2/8 = 1/4 = 0.25
3. P(A and B) - Probability that the selected number is both even and a multiple of 3:
The numbers that satisfy both conditions are 6. So, the favorable outcomes for event A and event B happening together are 1.
Total outcomes in the sample space = 8 (since there are 8 numbers).
P(A and B) = favorable outcomes / total outcomes = 1/8
4. P(B ┤| A) - Probability of event B given that event A has occurred:
Since event A has occurred (the number is even), the possible outcomes are 2, 4, 6, and 8. Out of these, only 6 is a multiple of 3. So, the favorable outcomes for event B given event A is 1.
Total outcomes in event A = 4 (since there are 4 even numbers).
P(B ┤| A) = favorable outcomes / total outcomes = 1/4 = 0.25
5. P(A ┤| B) - Probability of event A given that event B has occurred:
Since event B has occurred (the number is a multiple of 3), the possible outcomes are 3 and 6. Out of these, 6 is an even number. So, the favorable outcomes for event A given event B is 1.
Total outcomes in event B = 2 (since there are 2 multiples of 3).
P(A ┤| B) = favorable outcomes / total outcomes = 1/2 = 0.5
Now, to determine whether events A and B are independent:
Two events, A and B, are considered independent if P(B ┤| A) = P(B) and vice versa, P(A ┤| B) = P(A).
For event B given event A:
P(B ┤| A) = 0.25
P(B) = 0.25
The probabilities are equal, so event B is independent of event A.
For event A given event B:
P(A ┤| B) = 0.5
P(A) = 0.5
The probabilities are equal, so event A is independent of event B.
Therefore, events A and B are independent since the probability of each event is not affected by the occurrence of the other event.