Tutorial 3

Learning objectives

After this tutorial the student should be able to:

  • mention, explain and apply the three probability properties, the general addition rule, and the product rule for independent events;

  • recognize a random variable;

  • distinguish between a discrete probability distribution and a continuous probability distribution;

  • recall the properties of a binomial experiment;

  • mention and apply the formula of the expected outcome, the variance, and the standard deviation of a binomial distribution;

  • determine the probability of an event resulting from a binomial experiment using a binomial distribution table.

Probability laws

Read:

    • paragraph 4.1 pp.149-152 up to the abstract of the research question, and

    • paragraph 4.3 pp.155-158, or

    • paragraph 4.1 pp.140–143 up to the abstract of the research question, note: \(P(\mbox{even}\ E)\) should read \(P(\mbox{event}\ E)\) on p. 142, and

    • paragraph 4.3 pp.146-149.

The three probability properties are:

  • \(P(S) = 1\) where \(S\) is the sample space (set of all possible outcomes)

  • \(0 \leq P(A) \leq 1\) for any event \(A\)

  • The complement \(\bar{A}\) of an event consists of all outcomes that are not occurring in \(A\) implying \(P(A) = 1- P(\bar{A})\) (complement rule).

Two rules for probabilities are:

  • the general addition rule (in O&L referred to as ‘probability of union’):
    \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
    with the special case of mutually exclusive events \(A\) and \(B\):
    \(P(A \cup B) = P(A) + P(B)\).

  • the product rule for independent events: \(P(A \cap B) = P(A) \times P(B)\)

Random variables

(Re-)Read:

    • paragraph 4.6 pp.164-166, or

    • paragraph 4.6 pp.155–157,

where the definition of a random variable is given. A random variable can be discrete or continuous.

Random variables: discrete random variables

Read:

    • paragraph 4.7 pp.166-167, or

    • paragraph 4.7 pp.157–158,

where the term probability distribution is discussed. This is also called probability mass function.

Discrete probability distributions: the binomial distribution

Read:

    • paragraph 4.8 pp.167-175 up to Poisson distribution, or

    • paragraph 4.8 pp.158–166 up to Poisson distribution,

using the binomial distribution as an example of a discrete probability distribution.

The binomial distribution with parameters \(n\) and \(\pi\):

  • Assumptions for a binomial experiment:
    • \(n\) identical trials, where \(n\) denotes the sample size;
    • the \(n\) trials are independent of each other;
    • each trial results in one of two possible outcomes (one labelled as a success, and the other as a failure);
    • the probability of success in a single trial is denoted by \(\pi\), and the value of \(\pi\) is the same from trial to trial (constant);
    • the random variable \(y\) denotes the number of successes in \(n\) trials.
  • Notation: \(y \sim \mbox{Bin}(n, \pi)\mbox{ or }y \sim \mbox{B}(n, \pi)\)
  • Expected value: \(\mbox{E}(y) = \mu_y = n \times \pi\)
  • Variance: \(\mbox{var}(y) = \sigma^2_y = n \times \pi \times (1 - \pi)\) and standard deviation \(\sigma(y) = \sqrt{n \times \pi \times (1 - \pi)}\)
  • Probability distribution: \(P(y = k) = \frac{n!}{k! \times (n - k)!} \times \pi^k \times (1 - \pi)^{n - k}\ \forall\ k \in \{0, 1, 2,\ldots, n\}\)
ImportantRemarks about binomial probability calculation and shape of a binomial distribution.

  • \(n!\) is the notation for the factorial of \(n\), i.e., \(n! = n \times (n - 1) \times (n - 2) \times \ldots \times 3 \times 2 \times 1 = n \times (n - 1)!\). For example: \(4! = 4 \times 3 \times2 \times 1 = 24\)

  • For \(\pi = 0.5\) the binomial distribution is symmetric, see Figure 1 for a plot of an example.

Figure 1: Binomial distribution for \(n = 20\) and \(\pi = 0.5\)

Exercises to be done during the tutorial

Exercise 3.1 up to and including Exercise 3.4 are in the presentation handouts of Tutorial 3. For answers/feedback check Brightspace.

Exercises to be done after the tutorial

For answers/feedback check Brightspace.

Exercise 3.5

Do either

Exercise 3.6

Do either

Exercise 3.7

Do either

Exercise 3.8

Do either

Exercise 3.9

Do either

Exercise 3.10

Do either

Exercise 3.11

Do either