Scientific Reasoning

This required course is intended for all students in the SV Bachelor’s program. It introduces the main tools of discrete probability that are useful to biologists for understanding random phenomena, particularly those involving counting variables. The course is designed at a level accessible to students whose only prerequisite is a basic understanding of probability theory covered in the 10th grade of high school. The course uses concrete examples as a starting point to move toward modeling. 

• An introductory section covers the concept of sets, operations on sets, and the simple formalization of propositions.
• Part Two introduces probability terminology and reviews basic probability calculations (tables, trees) and conditional probabilities. The examples focus on real-world situations: calculating probabilities in a population stratified by age and gender, and diagnostic tests (sensitivity/specificity).
• Part 3 is devoted to a presentation of the main discrete probability models—the binomial, geometric, and Poisson models—and their applications. The concept of independent variables is introduced in a heuristic manner, with the goal of providing tools for calculating the expected value and variance of the sum of random variables.
• A few numerical simulations may be presented to illustrate the concept of fluctuation in a random variable or the convergence of the binomial distribution to the normal or Poisson distribution.

To provide the basic tools for calculating probabilities and using common discrete random variables in the context of their application to random phenomena in the life sciences.

10th-grade mathematics, Course HAV109X: Computational Methods

1) Sets

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

— Notation for the number sets N, Z, D, and Q.

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

2) Modeling Randomness: Calculating Probabilities

— The set (universe) of outcomes. Events. Union, intersection, complement.

— Probability distribution. Probability of an event: the sum of the probabilities of the possible outcomes. The relationship P(A∪B) + P(A∩B) = P(A) + P(B).

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

— Conditional probabilities and independence: the conditional probability of an event B given that an event A has a nonzero probability. Notation: PA(B).

— Independence of two events and mutual independence

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

— A sequence of independent trials, Bernoulli's model

3) Real-valued random variables 

— Real-valued random variable: modeling the numerical outcome of a random experiment; formalized as a function defined on the sample space and taking real-valued outputs.

— Distribution of a random variable. Expectation, variance, and standard deviation of a random variable

— Bernoulli's experiment, Bernoulli's principle

— 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 trial 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). (Assumed or can be proven for two discrete random variables on a finite sample space)

Additivity relation for independent variables X and Y: 

V(X+Y) = V(X) + V(Y). The relation V(aX) =^a²V(X).

Application to the expected value and variance of the binomial distribution.

— Other Specific Laws:  

Uniform distribution over {1, 2, …, n} 

Geometric distribution (rank of the first success in a sequence of independent Bernoulli trials): (assumed) expected value, memoryless property.

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

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

Hourly volumes*:

            CM: 12:00 p.m.

            TD: 9:00 p.m.

            Practical Work:

            Lot: