Statistics & Probability BS1103 — المسار الكامل

Random experiment

Recognize a random experiment and distinguish outcomes that are known from the outcome that actually occurs.

Sample space

Construct the complete sample space S for coin and die experiments.

Events and simple events

Interpret an event as a subset of S and identify simple events.

Equally likely probability rule

Compute probability as desired outcomes divided by total equally likely outcomes.

Complement of an event

Use P(A') = 1 - P(A) and interpret impossible/certain events.

Union and intersection

Translate 'A or B' and 'A and B' and apply the addition rule.

Disjoint events

Recognize mutually exclusive events and use their simplified union rule.

Independent events

Test independence using P(A∩B)=P(A)P(B).

Disjoint versus independent

Avoid confusing mutually exclusive events with independent events.

Two-dice sample spaces

Model distinguishable dice with 36 ordered outcomes.

At least, neither, but not

Translate probability wording into event operations before calculating.

Multi-step probability strategy

Move from givens to events, rule selection, substitution, and checking.

Conditional probability meaning

Interpret P(A|B) as probability after restricting attention to cases where B occurred.

Conditional probability formula

Apply the conditional probability definition using intersection and the conditioning event.

Independence through conditional probability

Connect independence with unchanged conditional probability.

Conditional sample spaces

Rebuild the relevant sample space after information is given.

Dice conditional probability

Solve conditional probability using ordered dice outcomes.

Urns without replacement

Track changing probabilities in sequential selections without replacement.

Partition of a sample space

Recognize mutually exclusive and exhaustive events forming a partition.

Total probability

Compute an event probability across partition branches.

Bayes theorem

Reverse a conditional probability using prior probabilities and likelihoods.

Probability trees

Organize prior and conditional probabilities in a branching tree.

False positives and posterior probability

Apply Bayes reasoning to diagnostic-test style problems.

Machine-source inference

Infer the likely source of an observed defective product using Bayes theorem.

Random variable

Map random experiment outcomes to numerical values.

Discrete random variable

Recognize countable random-variable values.

Probability mass function

Build and validate a discrete probability distribution.

Cumulative distribution function

Compute cumulative probabilities for a discrete random variable.

Expected value

Calculate and interpret the mathematical expectation.

Variance and standard deviation

Measure spread using variance and standard deviation.

Bernoulli process

Check repeated independent success/failure trial assumptions.

Binomial random variable

Identify number of successes in n trials.

Binomial probability formula

Use n, p, q and x correctly in the binomial formula.

Exactly x successes

Translate exactly into one binomial probability.

At least x successes

Translate at least into a tail sum or complement.

At most x successes

Translate at most into a cumulative binomial probability.

Poisson process intuition

Recognize random event counts occurring at a rate over an interval.

Poisson parameter

Interpret the Poisson mean/rate parameter.

Poisson probability formula

Calculate the probability of exactly x events.

Exactly x Poisson events

Compute a single Poisson probability.

At most x Poisson events

Sum Poisson probabilities through x.

Upper-tail Poisson events

Use sums or complements for upper-tail probabilities.

Continuous random variable

Distinguish continuous from discrete random variables.

Probability density function

Check nonnegativity and total area one for a PDF.

Point probability is zero

Understand why a single exact value has zero probability for a continuous variable.

Probability as area

Interpret interval probability as area under the density curve.

Continuous CDF

Build F(x) by integrating the density.

Interval probabilities from PDF or CDF

Compute probabilities over bounded and tail intervals.

Normal distribution

Recognize the bell-shaped normal model and its parameters.

Standard normal variable

Recognize the standard normal distribution.

Z-score transformation

Standardize X using its mean and standard deviation.

Normal-table meaning

Interpret the cumulative probability reported by the standard normal table.

Positive Z lookup

Locate a positive Z value in the table.

Negative Z by symmetry

Use symmetry to handle negative Z values.

Joint probability distribution

Represent simultaneous values of two discrete random variables.

Valid joint PMF

Check nonnegativity and total joint probability one.

Joint-event probability

Sum cells satisfying conditions on X and Y.

Marginal distribution of X

Sum joint probabilities over Y.

Marginal distribution of Y

Sum joint probabilities over X.

Independence of random variables

Check whether joint probabilities factor into marginals.

Population and sample

Distinguish a statistical population from a selected sample.

Parameter and statistic

Distinguish population characteristics from sample characteristics.

Ungrouped data

Recognize data in its original ungrouped form.

Frequency

Count how often a data value occurs.

Frequency table

Arrange observed values with their frequencies.

Reading a frequency distribution

Recover counts and totals from a frequency table.

Correlation concept

Describe correlation as strength and direction of association between two variables.

Positive and negative correlation

Recognize increasing-together and opposite-direction patterns.

Strength from scatterplots

Distinguish strong, weak, and no linear correlation visually.

Correlation is not regression

Distinguish measuring association from fitting a prediction line.

Pearson correlation coefficient

Calculate Pearson's correlation coefficient from paired observations.

Correlation range and sign

Interpret the coefficient's sign and allowable range.