# Probability and the additive rule

## Probability

Probability is the study of experiments. Experiments result in outcomes (also called simple events). A set or collection of outcomes is called an event. These concepts can be illustrated with a deck of cards. Pulling a card from the deck is an experiment. Getting the ace of spades is an outcome (or simple event) (there are 52 possible outcomes for the experiment of drawing a card from a deck. Events include getting a spade (which event contains 13 outcomes), or getting a king (which event contains 4 outcomes).

Probabilities are assigned to the outcomes of an experiment. We shall only consider experiments where all the outcomes are equally likely. Hence for drawing a card from a deck, each outcome has probability 1/52. The probability of an event is the sum of the probabilities of the outcomes in the event, hence the probability of drawing a spade is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13. Although the fact that the probability of getting a spade is 1/4 may make it seem that events are equally likely (since there are four suits), the probability of getting a face card is not equal to the probability of getting a number card. The only way to calculate the probability of an event is to sum the probabilities of the (equally likely) outcomes in the event. However, identifying what the equally likely outcomes are can be subtle as the following exercise illustrates.

Exercise: What are the equally likely outcomes for flipping a pair of coins? What are the equally likely outcomes for rolling a pair of dice?

## Additive rule

Since the the probability of an event is the sum of the probabilities of the outcomes which comprise the event, one might assume that the probability of an event is the sum of the probabilities of any events which comprise that event. However, The probability of getting a black card or an ace [which we may denote as P(black or ace)] is not P(black) + P(ace) since the former is 28/52 (there are 26 black cards and 2 red aces) while the latter is 26/52 + 4/52. The discrepancy is due to the fact that the black aces are counted twice on the right hand side, once with the black cards and once with the aces. Correcting for the double counting provides the additive rule for arbitrary events: P(A or B) = P(A) + P(B) - P(A and B). Indeed 28/52 = 26/52 + 4/52 - 2/52 (there are two black aces).

Definition. Two events A and B are said to be mutually exclusive (or disjoint) if their intersection is empty (or equivalently, P(A and B) = 0). For example, the events getting a club and getting a one-eyed jack are mutually excllusive because the one-eyed jacks are the jacks of spades and hearts. The events getting a heart and getting a one-eyed jack are not mutually exclusive, because the jack of hearts is both a heart and a one eyed-jack. The additve rule for mutually exclusive events is P(A or B) = P(A) + P(B) (because P(a and B) which should be subtracted from the right hand side is equal to zero).

Definiton: A' (read A complement) is the set of outcomes which are not in A. It follows that P(A or A') = 1 and P(A and A') = 0.

Competencies:WHat is P(black or 2) What is P(Black and 2)? If you fip 3 coins, what is the probability of getting two heads? If you roll two dice, what is the probability that the sum of the pips will be 5?

If P(A)=.4, P(B)=.3 and P(A and B)=.2, what is P(A or B)=?

If A is the set of one-eyed jacks, what is A'?

Reflection: Can you think of an experiment where the outcomes are not equally likely events?

Challenge:

May 2002