Rules of Probability Lesson

We can think about real life probability models like rolling a die or dice, or picking from a deck of cards. There are rules that govern mathematical calculations of probability. Application of these rules is conditional upon validating certain beginning assumptions about the individual trials. In order for random chance to be preserved it is necessary that the trials be independent from one another. When selecting cards from a deck randomly, each card will have the same chance for selection as long as repeated draws use the ENTIRE deck. That would be called repeated trials "with replacement" and would insure that the probability on each draw is the same. If the cards are not replaced then, logically, a different rule would be invoked. Without card replacement the probability for each subsequent draw would be different based on the number of cards left in the deck.

When considering multiple events it is important to know if one "prevents" the other from happening. An example would be you cannot receive a grade of both an A and a B in the same course for the same semester. We say those events are "mutually exclusive" or "disjoint" and represent a special situation.

Independence is another relationship that should be considered when working with two or more events. Two events are independent if the occurrence of one of the events does not affect the probability of the occurrence of the other event.

Common Sense Examples

Example: Tossing a coin and getting a head (event A) and then rolling a die and getting a 5 (event B).

Example: Rolling a die and getting a 6 (event A) and then rolling the die again and getting a 4 (event B).

Example: Selecting an ACE (event A) from a deck of cards, replacing it, and then selecting a two (event B) from a deck of cards.

Mathematically, there are three ways to check whether two events are independent. The notation P(A|B) is read as "probability of event A occurring GIVEN THAT event B has already occurred."

If P(A|B) = P(A) or if P(B|A) = P(B) then A and B are independent. These two statements are the mathematical version of knowing one event has occurred has NO EFFECT on the probability of the other.

It is also true that if P(A and B) = P(A) x P(B) then Events A and B are independent.

Another Example

Example—Consider the gender of two children in a family

Possible outcomes are MM, MF, FM, FF giving each outcome a probability of 1 out of 4.

Let A be the event that the first child is male

Let B be the event that the second child is female

P(A) =1/2 and P(B) =1/2

P(A and B) = ¼ from P(A) x P(B) = 1/2 x1/2 = ¼

Thus, A and B are independent events and agrees with our intuitive understanding of subsequent birth gender as unrelated to previous births.

For a complete explanation of the meaning of mutually exclusive and how it relates to the RULES for probability, please click on this sentence to download the explanations.

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