Two sets A and B are said to be independent if the occurrence of one does not affect the occurrence of the other.
How is the independence of two sets mathematically represented?
The independence of sets A and B is represented as P(A ∩ B) = P(A) * P(B)
What does P(A) represent in the formula P(A ∩ B) = P(A) * P(B)?
P(A) represents the probability of set A occurring.
What does P(B) represent in the formula P(A ∩ B) = P(A) * P(B)?
P(B) represents the probability of set B occurring.
If two sets A and B are independent, what can be said about their complements?
If A and B are independent, then the complement of A and the complement of B are also independent.
What is the general formula to calculate the probability of the union of two independent events A and B?
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
What is the probability of the intersection of two independent events A and B in terms of their individual probabilities?
P(A ∩ B) = P(A) * P(B)
True or False: If two events A and B are mutually exclusive, they are also independent.
True, if two events are mutually exclusive, they cannot occur together, so they are independent.
True or False: If two events A and B are independent, they are also mutually exclusive.
False, if two events are independent, they can occur together, so they are not mutually exclusive.
What is the conditional probability of event A given event B in terms of their individual probabilities?
P(A|B) = P(A) if A and B are independent
How can you prove the independence of two sets A and B using conditional probability?
If P(A|B) = P(A), then A and B are independent.
Give an example of two independent sets.
Example: Tossing a fair coin and rolling a fair die are independent events.
What is the probability of selecting a queen from a deck of cards and rolling a 2 with a fair die (independent events)?
P(selecting a queen) * P(rolling a 2) = 4/52 * 1/6 = 1/78
What is the probability of drawing two red cards (without replacement) from a standard deck of cards?
P(drawing first red card) * P(drawing second red card) = 26/52 * 25/51 = 25/102
In a bag, there are 5 red balls and 7 blue balls. If two balls are drawn without replacement, what is the probability of getting one red ball and one blue ball?
P(drawing one red ball) * P(drawing one blue ball) = (5/12) * (7/11) = 35/132
What is the definition of independent two sets?
Two sets A and B are said to be independent if the occurrence of one does not affect the occurrence of the other.
How is the independence of two sets mathematically represented?
The independence of sets A and B is represented as P(A ∩ B) = P(A) * P(B)
What does P(A) represent in the formula P(A ∩ B) = P(A) * P(B)?
P(A) represents the probability of set A occurring.
What does P(B) represent in the formula P(A ∩ B) = P(A) * P(B)?
P(B) represents the probability of set B occurring.
If two sets A and B are independent, what can be said about their complements?
If A and B are independent, then the complement of A and the complement of B are also independent.
What is the general formula to calculate the probability of the union of two independent events A and B?
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
What is the probability of the intersection of two independent events A and B in terms of their individual probabilities?
P(A ∩ B) = P(A) * P(B)
True or False: If two events A and B are mutually exclusive, they are also independent.
True, if two events are mutually exclusive, they cannot occur together, so they are independent.
True or False: If two events A and B are independent, they are also mutually exclusive.
False, if two events are independent, they can occur together, so they are not mutually exclusive.
What is the conditional probability of event A given event B in terms of their individual probabilities?
P(A|B) = P(A) if A and B are independent
How can you prove the independence of two sets A and B using conditional probability?
If P(A|B) = P(A), then A and B are independent.
Give an example of two independent sets.
Example: Tossing a fair coin and rolling a fair die are independent events.
What is the probability of selecting a queen from a deck of cards and rolling a 2 with a fair die (independent events)?
P(selecting a queen) * P(rolling a 2) = 4/52 * 1/6 = 1/78
What is the probability of drawing two red cards (without replacement) from a standard deck of cards?
P(drawing first red card) * P(drawing second red card) = 26/52 * 25/51 = 25/102
In a bag, there are 5 red balls and 7 blue balls. If two balls are drawn without replacement, what is the probability of getting one red ball and one blue ball?
P(drawing one red ball) * P(drawing one blue ball) = (5/12) * (7/11) = 35/132