Basic Definition

Elementary outcomes and events

• We perform an experiment, the cross Aa X Aa.
  An experiment is the process of observing a phenomenon that has variation in its outcome.
• We are interested in the offspring genotype: AA, Aa, or aa. These represent the elementary outcomes of the experiment. The elementary outcomes are mutually exclusive.
• An event is a set of elementary outcomes. E.g. The offspring has at least one copy of the A allele: this event consists of two elementary outcomes (AA and Aa).
• Each event has an associated probability
  One notation of probaility of an event is the frequency with which the event occured among an infinite number of experiments (trials).

Algebra of probabilities

``Or'' = addition rule

• Probability of an event with two or more mutually exclusive elementary outcomes.
• Add the probability of each elementary outcomes
• E.g. Probability that the offspring is AA or Aa is $Pr(AA)+Pr(Aa) = 1/4 + 1/2 = 3/4$

``And'' = multiplication rule

• Probability of event that two or more independent events occur simultaneously
• Multiply the probability of each event
• E.g. cross of Aa x AA. Aa can produce gamete A or a with probabilities Pr(A from Aa) = 1/2 and Pr(a from Aa) = 1/2. AA always produce gamete A, so Pr(A from AA) = 1 and Pr(a from AA) = 0. In this context, elementary outcomes are A or a. Probability of the event chossing two independent elemenatry outcomes (gamete A from Aa and gamete A from AA) is 1/2 * 1 = 1/2.

Examples

• The following examples are very simple. Notice that you are unconsciously using the addition and multiplication rules.

• Throw a fair die
  • What are the elementary outcomes?
  • What is the probability that the number is 2 or 4
• Throw a fair die twice
  • Probability that the first number is 2 or 4 and the 2nd number is even (i.e. 2 or 4 or 6)
  • Probability that the 2 numbers are equal
  • Probability that the 2 numbers are different

Practice

Calculate the following probablities:

1. 2 people have the same birthday. Assume 365 days/year, birthdays randomly distributed over a year.

2. 3 people have the same birthday.

3. All 3 people have the different birthday.

4. In a group of 3 people, at least one pair has the same birthday.

5. In a group of 20 people, at least one pair has the same birthday.

6. Throw 2 dice simultaneously. What is the probability that the product of these 2 values are 6?

7. Throw 2 dice simultaneously. Probability that one value is greater than the other.

8. If two dice are rolled three times, what is the probability that the two dice will match (i.e., display the same number) on one of the three rolls?

There are many examples when you can calculate probabilities from your daily life. Probability is not ``intuitive'' to most people, but it is something you can learn with more practices (and it's fun to figure out tricky probabilities). View the problems from multiple directions, consider all possible outcomes, and assign the probability of each outcome by reducing the problem to simple relationship of ``and'' and ``or''.