Probability: Axiomatic Approach and Theorems

A researcher is analyzing the probability of specific genes being expressed in a group of cells. What fundamental concept describes the set of all possible gene expression outcomes?

Which type of events cannot occur at the same time?

What is the primary difference between the classical and empirical approaches to probability?

In probability theory, what does the axiomatic approach primarily provide?

A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What concept is essential for calculating the probability of drawing a red ball followed by a blue ball?

A software company tests two different features independently. Feature A has a 95% chance of working, and feature B has a 90% chance. Assuming their performance is independent, what theorem helps to calculate the probability that both features work correctly in a system?

If two events are considered independent, what does this imply about their probabilities?

In a medical diagnosis scenario, what is Bayes' theorem primarily used for?

What is the significance of 'pairwise independence' compared to 'mutual independence' for a set of events?

Which of the following scenarios best illustrates a 'trial' in the context of probability?

What is the primary purpose of Boole's inequality in probability theory?

In the context of probability, what distinguishes 'exhaustive events' from other types of events?

A geneticist is studying the inheritance of two genes. Gene A is inherited in 80% of offspring, and Gene B is inherited in 60% of offspring. If these genes are inherited independently, what approach could be used to determine the chance they are both inherited?

In a manufacturing process, a machine produces items, and each item has a certain probability of being defective. What probabilistic concept is applied when assessing whether the defect rate on one machine affects the defect rate on another machine?

A teacher wants to predict the probability of a student passing both a math and science test. What probabilistic approach is appropriate if the teacher believes high performance in math increases the chance of high performance in science?

When is the addition theorem of probability most applicable?

In the context of random experiments, what is the significance of defining a 'sample space'?

In a clinical trial, researchers want to determine the probability that a new drug is effective, given some observed side effects. What tool would they use?

What is a limitation of the classical approach to probability?

If events A and B are mutually exclusive, what does this imply about $P(A \cap B)$ (the probability of both A and B occurring)?

A survey finds that 60% of people like coffee, 40% like tea, and 20% like both. What theorem do we use to determine the probability that a person likes either coffee or tea?

What condition must be met to directly apply the simplified form of the multiplication theorem $P(A \cap B) = P(A) * P(B)$

How does the empirical approach to probability differ from the theoretical approach in determining the likelihood of an event?

Why is the sample space a fundamental concept in probability?

What is the key distinction between 'pairwise independence' and 'mutual independence' of events?

What does the term 'event' represent in basic probability theory?

With regards to independence, how does mutual independence differ from pairwise independence?

How do you describe the difference between mutually exclusive events and independent events?

A new medicine is tested where it has a chance to create side effects. Given observed side effects, what technique will the researchers use to check the medicine's effectiveness?

Why do most real-world probabilistic problems utilize the empirical approach rather than classical?

To compute the likelihood of the union of events, what does Boole's inequality provide?

What’s the primary constraint when determining the probability of events using the classical method?

A team has an 80% chance to get a deal A and a 70% chance to get a deal B. Assuming they are independent, how do you find the chance that they get deal A and deal B?

When do you use the multiplication theorem considering conditional probability?

What makes exhaustive events unique?

How do we describe 'mutually exclusive events?'