Probability : 11th grade math lesson to download in PDF.

11th grade math lessons Report an error on this Mathovore page.Report an error / Note? 11th grade
Probability in a 11th grade math lesson where we will study the law of large numbers and the law of probabilities.
In this first grade lesson, we will also cover the concept of equiprobability, the expectation and standard deviation and the variance of a random variable.

I. Random variables and probabilities

Definition: discrete random variable.

Let \Omega\,=\{e_1,e_2,e_3,...,e_m\} be the finite universe of a random experiment.

A random variable X on \Omega is a function which, to each outcome of \Omega, associates a real number.

Definition: probability law of a discrete variable.

Let X be a random variable taking the values \{x_1,x_2,x_3,...,x_n\}. If for each value x_i, we associate the probability of the event (X=x_i), we define the probability law X.


The probability law of a random variable is presented in the form of a table.

We have P(X=x_1)+P(X=x_2)+P(X=x_3)+...+P(X=x_1)=1.

Random variables and probabilities

II. Expectation, variance and standard deviation.


Let X be a random variable taking the values x_1,x_2,x_3,...,x_n with the respective probabilities p_1,p_2,p_3,...,p_n.

  • We call the expectation of X the number E(x)=p_1x_1+p_2x_2+p_3x_3+...+p_nx_n=\sum_{i=1}^{n}p_ix_i.
  • We call the variance of X the number V(x)=p_1(x_1-E(X))^2+p_2(x_2-E(X))^2+p_3(x_3-E(X))^2+...+p_n(x_n-E(X))^2=\sum_{i=1}^{n}p_i(x_i-E(X))^2
  • We call the standard deviation of X the number \sigma\,(X)=\sqrt{V(X)}.

III. Affine transformation of a random variable.


Let X be a random variable taking the values x_1,x_2,x_3,...,x_n. For all real a and b, we can define another random variable, by associating to each outcome

giving the value x_i, the number ax_i+b.

We note this random variable aX+b.

  • E(aX+b)=aE(x)+b.
  • V(aX)=a^2V(X).

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