Scientific reasoning

This compulsory course is intended for all students enrolled in the Bachelor's degree program in Life Sciences. It presents the main tools of discrete probability that are useful to biologists for understanding random phenomena involving counting variables in particular. The course is designed to be accessible to students who have only studied the basics of probability covered in high school. The course uses concrete examples to introduce modeling. 

• A preliminary section introduces the concept of sets, operations on sets, and the simple formalization of propositions.
• The second part introduces probability vocabulary and covers basic probability calculations (tables, trees) and conditional probabilities. The examples relate to real-life situations: calculating probabilities in a population stratified by age and gender, diagnostic tests (sensitivity/specificity).
• The third part is devoted to presenting the main discrete probability models: binomial, geometric, Poisson, and their applications. The concept of independent variables is presented heuristically, with the aim of providing tools for calculating the expected value and variance of the sum of random variables.
• Some numerical simulations may be presented to illustrate the concept of fluctuation of a random variable or the convergence of the binomial distribution to the normal distribution or Poisson distribution.

 

Provide the basic tools for calculating probabilities and using common discrete random variables in the context of random phenomena in the life sciences.

Second-year mathematics, EU HAV109X Computational methods

1) Sets

— Concepts of elements of a set, subsets, membership and inclusion, union, intersection, and complement, and knowing how to use the corresponding basic symbols: ∈, ⊂, ∩, ∪.

— Notation of the number sets N, Z, D, Q.

— Negation of simple propositions (without implication or quantifiers); counterexamples to show that a proposition is false; formulating an implication, a logical equivalence; reciprocal of an implication: simple set-theoretic examples.

2) Modeling chance: calculating probabilities

— Set (universe) of outcomes. Events. Meeting, intersection, complementary.

— Probability law (or distribution). Probability of an event: sum of the probabilities of the outcomes. Relationship P(A∪B)+P(A∩B) =P(A)+P(B).

— Counting using tables and trees (product rule, sum rule).

— Conditional probabilities and independence: conditional probability of event B given event A with non-zero probability. Notation PA(B).

— Independence of two events and mutual independence

— Partition of the universe (complete event systems). Total probability formula. Bayes' formula 

— Sequence of independent trials, Bernoulli diagram

 

3) Real random variables 

— Real random variable: modeling the numerical result of a random experiment; formalization as a function defined on the universe and with real values.

— Law of a random variable. Expectation, variance, standard deviation of a random variable

— Bernoulli's experiment, Bernoulli's law

— Binomial distribution B(n,p): distribution of the number of successes. Expression using binomial coefficients. Binomial coefficients: definition (number of ways to obtain k successes in a Bernoulli scheme of size n), Pascal's triangle.

— Sums of random variables

Linearity of expectation: E(X+Y) = E(X) + E(Y) and E(aX) = aE(X). (accepted or can be demonstrated for two discrete random variables on a finite universe)

Additivity relationship for independent variables X, Y: 

V(X+Y) = V(X) + V(Y). Relationship V(aX) =^a²V(X).

Application to expectation, variance of the binomial distribution.

— Other discrete laws:  

Uniform law on {1,2,…, n} 

Geometric law (rank of the first success in a sequence of independent Bernoulli trials): expectation (admitted), memoryless law property.

Poisson distribution: characteristics and properties (results on accepted series), approximation of the binomial distribution by the Poisson distribution (case of rare events).

 Computational examples: pbinom, rbinom, rpois, ppoiss, rgeom, pgeom functions, illustration/popularization of the law of large numbers: convergence of the proportion of event A in a sample of size n towards probability P(A).

Hourly volumes:

            CM: 12 p.m.

            TD: 9 p.m.

            TP:

            Land: