Can you apply the multiplication rule of probability with the addition rule of probability? Please explain your answer in one or two sentences or give an example.
Answers
Answer:
The Multiplication Rule
If A and B are two events defined on a sample space, then: P(A AND B) = P(B)P(A|B).
This rule may also be written as
P
(
A
∣
B
)
=
P
(
A
AND
B
)
P
(
B
)
(The probability of A given B equals the probability of A and B divided by the probability of B.)
If A and B are independent, then P(A|B) = P(A). Then P(A AND B) = P(A|B)P(B) becomes P(A AND B) = P(A)P(B).
The Addition Rule
If A and B are defined on a sample space, then: P(A OR B) = P(A) + P(B) – P(A AND B).
If A and B are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) – P(A AND B) becomes P(A OR B) = P(A) + P(B).
EXAMPLE
Klaus is trying to choose where to go on vacation. His two choices are: A = New Zealand and B = Alaska
Klaus can only afford one vacation. The probability that he chooses A is P(A) = 0.6 and the probability that he chooses B isP(B) = 0.35.
P(A AND B) = 0 because Klaus can only afford to take one vacation
Therefore, the probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) = 0.6 + 0.35 = 0.95. Note that the probability that he does not choose to go anywhere on vacation must be 0.05.
EXAMPLE
Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal GIVEN that he made the first goal is 0.90.
What is the probability that he makes both goals?
What is the probability that Carlos makes either the first goal or the second goal?
Are A and B independent?
Are Aand B mutually exclusive?
Solution:
The problem is asking you to find P(A AND B) = P(B AND A). Since P(B|A) = 0.90: P(B AND A) = P(B|A) P(A) = (0.90)(0.65) = 0.585Carlos makes the first and second goals with probability 0.585.
The problem is asking you to find P(A OR B).
P(A OR B) = P(A) + P(B) – P(A AND B) = 0.65 + 0.65 – 0.585 = 0.715Carlos makes either the first goal or the second goal with probability 0.715.
No, they are not, because P(B AND A) = 0.585.
P(B)P(A) = (0.65)(0.65) = 0.4230.423 ≠ 0.585 = P(B AND A)So, P(B AND A) is not equal to P(B)P(A).
No, they are not because P(A and B) = 0.585. To be mutually exclusive, P(A AND B) must equal zero.
Answer:
yes the example 20×2=40