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If events A and B are independent, and $P(A) = 0.4$ and $P(B) = 0.6$, what is the probability of both A and B occurring?

Given two dependent events, where $P(X) = 0.5$ and $P(Y|X) = 0.8$, what is $P(X \text{ and } Y)$?

If events A and B are mutually exclusive, and $P(A) = 0.3$ and $P(B) = 0.4$, what is the probability of either A or B occurring?

Events R and S are not mutually exclusive. If $P(R) = 0.6$, $P(S) = 0.5$, and $P(R \text{ and } S) = 0.2$, what is $P(R \text{ or } S)$?

Given $P(E) = 0.5$ and $P(F|E) = 0.3$, calculate $P(E \text{ and } F)$.

In a Venn diagram, if set A represents prime numbers less than 10 and set B represents odd numbers less than 10, what does $A \cap B$ represent?

Using Venn diagrams, determine which formula correctly calculates the probability of the union of three events ($A \cup B \cup C$).

If $P(X) = 0.7$, what is $P(X')$ using the complement rule?

In the context of tree diagrams, what does each branch represent?

When using a tree diagram for a sequence of two dependent events, how do you calculate the probability of a specific path?

Consider a tree diagram where the first event has two outcomes with probabilities 0.4 and 0.6. From the 0.4 outcome, the second event has probabilities 0.7 and 0.3. What is the probability of following the path with probabilities 0.4 and then 0.3?

In a contingency table, what does the marginal total represent?

Given a contingency table, if $n(A \text{ and } B) = 20$, $n(A) = 50$, and $n(B) = 40$, with $n(\text{Total}) = 100$, are events A and B independent?

In conditional probability, what does $P(A|B)$ represent?

Given $P(A) = 0.6$, $P(B) = 0.8$ and $P(A \cap B) = 0.5$, what is $P(B|A)$?

If events A and B are mutually exclusive, what can you say about $P(A \cap B)$?

You have a bag with 5 red balls and 3 blue balls. You draw one ball, do not replace it, and then draw another. What is the probability that both balls are red?

Consider events A, B, and C. If A and B are independent, B and C are mutually exclusive, and A and C cover the entire sample space, and given that $P(A) = x$, $P(B) = y$, derive an expression for P(C).

Imagine a scenario where events A and B are neither independent nor mutually exclusive. Given $P(A) > 0$, $P(B) > 0$, $P(A) = 0.7$, and $P(A \cup B) = 0.9$, is it possible to uniquely determine the value of $P(B)$ without additional information, and if so, what is its value?

Consider a game where you flip a biased coin twice. The probability of heads is $p$, with $0 < p < 1$. Let A be the event 'first flip is heads,' and B be the event 'both flips are the same.' Determine the condition on p for which A and B are independent.

If event J and event K are independent, which equation correctly describes the probability of both events J and K occurring?

For two events, X and Y, to be considered mutually exclusive, what must be true about their ability to occur simultaneously?

Given events M and N are not mutually exclusive, and you know $P(M)$ and $P(N)$, which additional probability do you need to calculate $P(M ext{ or } N)$?

In a Venn diagram, what does the region representing $A \cap B$ signify?

Using Venn diagrams, which formula correctly calculates the probability of the union of two events ($A \cup B$) that are NOT mutually exclusive?

If $P(G) = 0.25$, what is the probability of the complement of G, denoted as $P(G')$?

In a tree diagram, what does each branch point represent?

When using a tree diagram for dependent events, how are the probabilities on the second set of branches determined?

Consider a two-stage experiment represented by a tree diagram. To find the probability of a specific sequence of outcomes, you should:

In a contingency table, what do marginal totals represent?

Given a contingency table, if you want to calculate $P(A|B)$, which counts are needed?

If events R and S are independent, how should the probability of their intersection $P(R \cap S)$ relate to their individual probabilities $P(R)$ and $P(S)$?

Consider two events, E and F, where $P(E) = 0.6$ and $P(F|E) = 0.4$. What is the probability of both E and F occurring, $P(E ext{ and } F)$?

Events X and Y are mutually exclusive. If $P(X) = 0.4$ and $P(Y) = 0.5$, what is the probability of either X or Y occurring, $P(X ext{ or } Y)$?

Events A and B are such that $P(A) = 0.7$, $P(B) = 0.3$, and $P(A \cap B) = 0.2$. What is the conditional probability $P(A|B)$?

If two events are mutually exclusive, what can be definitively stated about their intersection?

Consider three events A, B, and C. Which term in the general addition rule for $P(A \cup B \cup C)$ corrects for overcounting the outcomes that are in the intersection of all three events?

In the context of contingency tables and independence testing, if events A and B are independent, what relationship should hold between $P(A ext{ and } B)$, $P(A)$, and $P(B)$?

You are using a tree diagram to model a scenario with two sequential events. The first event has outcomes $O_1$ and $O_2$ with probabilities $p_1$ and $p_2$ respectively. From $O_1$, the second event has outcomes $O_{1a}$ and $O_{1b}$ with conditional probabilities $p_{1a}$ and $p_{1b}$. What is the probability of the sequence of outcomes $O_1$ followed by $O_{1a}$?

Consider events D, E, and F. Events D and E are mutually exclusive. Which simplification can be made to the general addition rule $P(D \cup E \cup F) = P(D) + P(E) + P(F) - P(D \cap E) - P(D \cap F) - P(E \cap F) + P(D \cap E \cap F)$?

If two events are described as 'independent', what does this imply about how the occurrence of one event affects the probability of the other?

For dependent events A and B, which formula correctly calculates the probability of both events A and B occurring, $P(A ext{ and } B)$?

What is the defining characteristic of mutually exclusive events?

Which formula is used to calculate the probability of either event A or event B occurring when A and B are NOT mutually exclusive?

In the context of conditional probability, $P(A|B)$ is read as 'the probability of A given B'. What does the condition '| B' imply?

In a Venn diagram, what does the region representing $A \cap B$ (A intersection B) signify?

Using Venn diagrams, which set operation represents the outcomes that are in event A or in event B or in both?

What does the complement of an event A, denoted as $A'$, represent in probability?

Consider a tree diagram used to represent sequential events. What does each 'branch point' in the diagram signify?

In a tree diagram, how is the probability of a specific path calculated?

For events A and B to be considered independent based on a contingency table, what condition must be met in terms of their probabilities or counts?

Given the general addition rule for three events $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$, why is the term $P(A \cap B \cap C)$ added?

If events D and E are mutually exclusive, what simplification occurs in the general addition rule $P(D \cup E \cup F) = P(D) + P(E) + P(F) - P(D \cap E) - P(D \cap F) - P(E \cap F) + P(D \cap E \cap F)$?

Consider a scenario with two sequential events represented by a tree diagram. The first event has outcomes $O_1$ and $O_2$. From $O_1$, the second event has outcomes $O_{1a}$ and $O_{1b}$. What probability is needed to determine the probability of the path leading to $O_1$ followed by $O_{1a}$?

If event J and event K are independent, which of the following equations MUST be true?

Given events R and S are not mutually exclusive. To calculate $P(R ext{ or } S)$, what minimal set of probabilities do you need to know?

In a contingency table designed to analyze the relationship between event X (with categories X1, X2) and event Y (with categories Y1, Y2), if you want to calculate the conditional probability $P(X1|Y2)$, which counts are directly needed from the table?

Given that events A and B are mutually exclusive, what is the value of $P(A \cap B)$?

Consider a scenario where you draw two balls without replacement from a bag. Are the events 'the first ball is red' and 'the second ball is red' independent or dependent events?