Chapter 4 Definitions p3 Flashcards
(13 cards)
Independent Variable
Independent Variable
In statistics, an independent variable is the variable that is controlled or chosen to observe its effect on another variable. It is often denoted as the input or cause in an experiment or data analysis. In probability, the term is less commonly used, but in statistical modelling (like regression), it refers to the predictor variable. It does not depend on the value of any other variable in the study. For example, time or dosage in a drug trial is usually the independent variable.
Dependent Variable
Dependent Variable
The dependent variable is the outcome or response that is measured and is expected to change as a result of changes in the independent variable. It depends on the input or cause being tested. In statistical modelling, it is the variable we are trying to predict or explain. For instance, in an experiment to test how sunlight affects plant growth, plant height is the dependent variable. In graphs, it is usually plotted on the vertical (y) axis.
sample space diagram
A sample space diagram is typically presented as a grid or table, especially when two events (like rolling two dice) are involved. Each cell in the diagram represents one outcome from combining the results of both events.
🔹 Example: Tossing two six-sided dice
1 2 3 4 5 6
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3 5,4 5,5 5,6
6 6,1 6,2 6,3 6,4 6,5 6,6
The rows represent outcomes from the first die, and the columns represent outcomes from the second die. Each cell, such as (3,4) or (6,2), shows one possible combined result. This creates a total of 36 possible outcomes, so the sample space S is:
S = {(1,1), (1,2), …, (6,6)}
This setup is useful to calculate probabilities, for example, the probability of the sum being 7, to identify favourable outcomes, and to visualise compound events. It is especially helpful for independent events and situations where all outcomes are equally likely.
Conditional probability:
Conditional probability: The probability of an event occurring given that another event has already occurred. Written as P(A | B) = P(A ∩ B) / P(B). It uses a reduced sample space where the condition B is known to have happened. It measures how the likelihood of A changes when B is true. Used when events are not independent.
Modelled:
Modelled: Represented a real-world situation using mathematical or statistical methods. In statistics, it means using data to describe patterns, trends, or relationships. A model simplifies reality to make predictions or draw conclusions. It can be visual (like a graph) or algebraic (like an equation). Assumptions are often made to keep the model workable.
P(B | A)
P(B | A) means the probability of event B occurring given that event A has already occurred. It tells you how likely B is when you know A is true. The formula to calculate this is:
P(B | A) = P(A ∩ B) / P(A)
This is used when the events A and B are not independent, and you want to update the probability of B based on the fact that A has occurred.
what does this mean? P(A|B)=P(A|B’)=P(A), and P(B|A)=P(B|A’)=P(B)
This notation expresses the idea of independence between events A and B.
P(A | B) = P(A | B′) = P(A) means:
The probability of A is not affected by whether B happens or not. So, A is independent of B.
P(B | A) = P(B | A′) = P(B) means:
The probability of B is not affected by whether A happens or not. So, B is independent of A.
In summary:
If knowing whether B happens (or not) doesn’t change the probability of A, and vice versa, then A and B are statistically independent.
Restricted sample space:
Restricted sample space: A smaller set of possible outcomes based on a given condition. When calculating conditional probability, you only consider outcomes where the condition is true. This restriction narrows the original sample space. For example, if you know event B has occurred, you only focus on outcomes where B happens. It helps adjust probabilities to reflect new information.
What does it mean to restrict the sampe space
To restrict the sample space means to focus only on the outcomes that satisfy a given condition, instead of considering all possible outcomes. This is usually done when calculating conditional probability. You ignore everything outside the condition and treat the condition as the new “universe” of outcomes. For example, if event B has occurred, you only look at outcomes where B is true. This helps you find probabilities relative to the condition, not the full original context.
Name all the types of of probability formulae.
Probability formulae in Venn diagrams:
Union (A or B):
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Intersection (A and B):
P(A ∩ B) = P(A) × P(B | A) if dependent
P(A ∩ B) = P(A) × P(B) if independent
Complement (not A):
P(A′) = 1 − P(A)
Mutually exclusive events (no overlap):
P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)
Total probability (everything in the diagram):
P(A) + P(A′) = 1
Addition formula:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Calculates the probability of A or B or both occurring. Subtracts the intersection to avoid double-counting. If A and B are mutually exclusive, then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
Multiplication formula:
Multiplication formula:
P(A ∩ B) = P(A) × P(B | A)
Calculates the probability of both A and B occurring. Used when events are dependent.
If A and B are independent, then P(A ∩ B) = P(A) × P(B).
Tree diagram:
Tree diagram:
A tree diagram is a visual tool used to map out all possible outcomes of a sequence of events. It shows branches for each possible event at each stage, making it easier to calculate combined probabilities. Each branch is labelled with the probability of that event occurring. Tree diagrams help organise complex probability problems, especially those involving conditional or sequential events. They allow you to multiply probabilities along branches to find overall probabilities.
