4.4: Conditional Probability and Independence Flashcards
What is conditional probability?
Conditional probability P(A | B) is the probability of event A occurring given that event B has already occurred.
It’s calculated as P(A ∩ B) / P(B), provided P(B) > 0.
How do you calculate the conditional probability of event A given event B?
To find P(A | B), divide the probability of both A and B occurring by the probability of B. Use the formula
P(A | B) = P(A ∩ B) / P(B).
How do you calculate the conditional probability of event B given event A?
To find P(B | A), divide the probability of both A and B occurring by the probability of A. Use the formula
P(B | A) = P(A ∩ B) / P(A).
How do you find the probability that a selected cable passing has Crystal’s cable television service given it has the internet service?
Use the formula for conditional probability: P(A | B) = P(A ∩ B) / P(B). If 6.5 million out of 9.8 million cable internet customers also have TV service, P(A | B) = 6.5 / 9.8 = 0.66.
How do you calculate the joint probability P(A ∩ B)?
The joint probability P(A ∩ B) can be calculated using the general multiplication rule:
P(A ∩ B) = P(A)P(B | A)
or
P(A ∩ B) = P(B)P(A | B).
What is the general multiplication rule for probabilities?
The general multiplication rule states that for any two events A and B, the probability of both A and B occurring is given by
P(A ∩ B) = P(A)P(B | A)
or
P(A ∩ B) = P(B)P(A | B).
How can conditional probabilities be used to calculate joint probabilities?
To calculate the joint probability P(A ∩ B), you can use conditional probabilities by multiplying
P(A) by P(B | A),
or
P(B) by P(A | B).
How do you find P(MGT ∩ W), the probability of selecting a sales representative who has a management position and is a woman?
P(MGT ∩ W) can be calculated using the general multiplication rule:
P(MGT ∩ W) = P(MGT)P(W | MGT).
If P(MGT) = 0.25
and
P(W | MGT) = 0.44,
then
P(MGT ∩ W) = 0.25 * 0.44 = 0.11.
How do you find P(MGT | W), the probability of a female sales representative having a management position?
P(MGT | W) is found by dividing P(MGT ∩ W) by P(W):
P(MGT | W) = P(MGT ∩ W) / P(W).
If P(MGT ∩ W) = 0.11 and P(W) = 0.52, then P(MGT | W) = 0.11 / 0.52 ≈ 0.
What defines two events A and B as independent?
Two events A and B are independent if
P(A | B) = P(A) and P(B | A) = P(B).
This means the occurrence of one event does not affect the probability of the occurrence of the other event.
How do you check if events MGT and W are independent?
To check if MGT and W are independent, compare P(MGT | W) with P(MGT).
If they are equal, the events are independent. If not, they are dependent.
What is the multiplication rule for two independent events?
If A and B are independent events, then P(A ∩ B) = P(A)P(B).
This rule simplifies the calculation of the joint probability when events do not influence each other.
How do you calculate the probability of two independent events occurring together?
For independent events A and B, the probability that both will occur is found by multiplying their individual probabilities:
P(A ∩ B) = P(A)P(B).
What is the multiplication rule for N independent events?
If A1, A2, …, AN are independent events, then the probability that all of these events will simultaneously occur is
P(A1 ∩ A2 ∩ … ∩ AN) = P(A1)P(A2) … P(AN).
What assumption can be made about customer satisfaction with different aspects of service?
It is often assumed that customer satisfaction with different aspects of service are independent events, meaning the satisfaction in one aspect does not influence satisfaction in another.
