Scientific reasoning

This compulsory course is aimed at all students in the SV License. It introduces the main tools of discrete probability which are useful to the biologist in understanding random phenomena involving counting variables. The course is set at a level accessible to students with only the basics of probability calculus covered in the second year of high school. The course focuses on concrete examples, leading on to modeling. 

• A preliminary section introduces the notion of sets, operations on sets and the simple formalization of propositions.
• The second part introduces the vocabulary of probability and covers elementary probability calculations (tables, trees) and conditional probabilities. The examples are based on real-life situations: calculating probabilities in a population stratified by age or gender, diagnostic tests (sensitivity/specificity), etc.
• The third part is devoted to the presentation of the main discrete law models: binomial, geometric, poisson and their applications. The notion of independent variables is presented in a heuristic way, the aim being to provide tools for calculating the expectation and variance of the sum of random variables.
• A few numerical simulations can be presented to illustrate the notion of fluctuation of a random variable or the convergence of the binomial distribution to the normal distribution or the poisson distribution.

Provide the basic tools for calculating probabilities and using the usual discrete random variables in a context of application to random phenomena from the life sciences.

second level mathematics, UE HAV109X Calculation methods

1) Sets

- Notions of element of a set, subset, membership and inclusion, meeting, intersection and complementary and know how to use the corresponding basic symbols: ∈,⊂,∩,∪.

- Notation of 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 examples.

2) Modelling chance: calculating probabilities

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

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

- Enumeration using tables and trees (product and sum rules).

- Conditional probability and independence: conditional probability of an event B knowing an event A with non-zero probability. Notation PA(B).

- Independence of two events and mutual independence

- Partitioning the universe (complete systems of events). Total probability formula. Bayes formula 

- Succession of independent events, Bernoulli scheme

3) Real random variables 

- Real random variable: modeling the numerical outcome 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 test, Bernoulli's law

- Binomial law B(n,p): law 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.

- Sum of random variables

Linearity of expectation: E(X+Y) =E(X)+E(Y) and E(aX) =aE(X). (admitted or can be demonstrated for two discrete a.v.'s on a finite universe)

Additivity relation for independent variables X,Y : 

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

Application to the expectation and variance of the binomial distribution.

- Other Discrete Laws :  

Uniform law on {1,2,..., n} 

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

Poisson's law: characteristics and properties (results on admitted series), approximation of the binomial law by Poisson's law (case of rare events).

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

Hourly volumes* :

            CM: 12h

            TD: 21h

            TP :

            Terrain :