Let's address each question one at a time:
Question 1
Which of the following formulas is used to mathematically recognize and explain the conditional probability of event A, given event B?
Correct Answer:
\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]
This formula correctly represents the conditional probability of event A given event B.
Question 2
Which of the following represents a dependent event?
Correct Answer:
Picking 2 snacks from the pantry without looking and keeping the first one before selecting the second one.
This event is dependent because the outcome of the first selection affects the outcome of the second selection.
Question 3
Which of the following correctly interprets the probability of choosing 2 rare cards from the bin? Assume that the cards are pulled without replacement.
Correct Answer:
\[\frac{5}{100} \cdot \frac{4}{99}\]
This expression captures the probability of first choosing one rare card and then choosing another without replacement.
Question 4
What is \( P(brown|black) \)?
Correct Answer:
\[\frac{26}{100} \cdot \frac{47}{99}\]
This interpretation is correct if we consider the conditional probability of choosing brown given that black has already been chosen.
Question 5
What is the probability of hanging the blue painting first, and then the green?
Correct Answer:
\[\frac{1}{5} \cdot \frac{1}{4}\]
This probability calculation is correct because there are 5 paintings to choose from initially and then 4 remaining paintings after the first one has been hung.
If you have further questions or need explanations on any of the correct answers, feel free to ask!