H Flashcards
What is Bayes’ Theorem primarily used for?
To answer questions about the role of one event’s occurrence in relation to another event
Define independent events in probability.
If P(B | A) = P(B), then A and B are independent events
What is the formula for conditional probability P(B | A)?
P(B | A) = P(A ∩ B) / P(A)
What does it mean if an event has a probability of 1?
The event is called a certain or sure event
If the probability of an event is zero, what is it called?
An impossible event
What is the range of probability values for any event A in sample space S?
0 ≤ P(A) ≤ 1
What is the formula for the union of two events A and B?
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Fill in the blank: If a trial results in n exhaustive mutually exclusive and equally likely events, the probability of event A is _______.
m/n
What is the Total Probability Theorem used for?
To calculate the probability of an event when conditional probabilities are known
What is meant by a ‘prior probability’?
P(A) is called ‘A prior Probability’ because it exists before gaining any information from the experiment
Define ‘posterior probability’.
P(Ai | B) is called ‘Posterior probability’ determined after knowing the results of the experiment
What is a random variable?
A real number x connected with an outcome of a random experiment E
What characterizes a discrete random variable?
It takes at most a countable number of values
What is the probability mass function (pmf)?
The probability function associated with discrete random variables
What is a continuous random variable?
A random variable that can take all possible values between certain limits
What is the probability density function (pdf)?
The probability function associated with continuous random variables
Fill in the blank: The variance of a random variable X is defined as _______.
E[(X - µ)²]
What does ‘sensitivity’ refer to in the context of diagnostic tests?
The probability of a positive test result given the presence of the disease
What is the probability that a randomly chosen individual from a population has at least one mutation if 40% have a wing mutation, 20% have an eye mutation, and 12% have both?
P(at least one mutation) = P(wing) + P(eye) - P(both)
True or False: If two events A and B are mutually exclusive, they can occur at the same time.
False
What is the formula for the intersection of two events A and B?
P(A ∩ B) = P(A) P(B | A) or P(B) P(A | B)
What is the expected value (mean) of a random variable X?
E(X) = Σ [x * P(x)] for discrete; ∫ x * f(x) dx for continuous
What does the complement of an event A represent?
The event that A does not occur, denoted as A′
What is the definition of an event in probability?
An event is a subset A of the sample space S, representing a set of possible outcomes.
What are the two types of events based on their occurrence?
An event can either occur or not occur.
What is a simple or elementary event?
An event consisting of a single point of S.
What is the sure or certain event in probability?
The sample space S itself.
What is the impossible event in probability?
The empty set ∅.
What does A ∪ B represent in probability?
A ∪ B is the event ‘either A or B or both,’ called the union of A and B.
What does A ∩ B represent in probability?
A ∩ B is the event ‘both A and B,’ called the intersection of A and B.
What is the complement of an event A?
A′ is the event ‘not A’, and A′ = S - A.
What does A - B represent?
A - B = A ∩ B′ is the event ‘A but not B.’
What are mutually exclusive events?
Events are mutually exclusive if A ∩ B = ∅, meaning they cannot both occur.
What is a random experiment?
An experiment where results can vary from one performance to another under nearly identical conditions.
What is a sample space?
A set S that consists of all possible outcomes of a random experiment.
What is the sample space for flipping a coin?
{H, T}.
What is the sample space for rolling a die?
{1, 2, 3, 4, 5, 6}.
What is the sample space for playing a football match?
{Win, Draw, Lose}.
Fill in the blank: Each performance in a random experiment is called a _______.
[trial].
What is an outcome in the context of a random experiment?
The result of a trial in a random experiment, also known as an elementary event or sample point.
What are equally likely events?
Outcomes that have no reason to expect one in preference to the other.
List examples of random experiments.
- Flipping a coin
- Rolling a die
- Sexual intercourse for reproduction
- Gambling
- Surgical procedure
What is the focus of probability theory?
The analysis of phenomena that take place in indeterministic or random circumstances.
What does the theory of probability provide?
Mathematical models for real-world phenomena involving randomness.
