Elementary Probability

Part 3: Rules for Assigning and Calculating Probabilities

Here is the fundamental rule for assigning probabilities to outcomes in a sample space. It reflects the intuitive idea that every outcome has a non-negative probability (A value of 0 is allowed.) and exactly one of the outcomes must occur.

Rule: The probability of each outcome must be a non-negative number, and the sum of the probabilities of the possible outcomes of an experiment must be 1.

Note that it follows from this fundamental principle that the probability of an event is always a number between zero and 1.

1. Let An be the event that the sum of the faces showing after the roll of two fair dice is n. Calculate P(An) for n = 2 ... 12. (For example P(A3) is the probability that the faces showing sum to 3.) Why should that the sum of these 11 probabilities be 1? Check that this is the case.

2. Consider the experiment of flipping a coin three times. If we denote a head by H and a tail by T, we can list the 8 possible ordered outcomes as (H,H,H), (H,H,T)… each of which occurs with probability of 1/8. Finish listing the remaining members of the sample space. Calculate the probability of the following events:
   1. All three flips are heads.

   2. Exactly two flips are heads.

   3. The first flip is tails.

   4. At least one flip is heads.

   5. The heads and tails alternate.

Let A and B be two events from a given sample space. What is the probability of either A or B happening. (When we say "A or B" we mean "A or B or both.") Is this probability the same as P(A) + P(B)?

3. In Ms. Nelson’s homeroom at Weaver High School ,18 of her 30 students are taking a history class and 14 are taking a biology class.
   1. If Ms. Nelson chooses a homeroom student at random, what is the probability that the student is taking a history class?

   2. If Ms. Nelson chooses a homeroom student at random, what is the probability that the student is taking a biology class?

   3. Explain why the sum of your answers from parts a) and b) is obviously not the probability that Ms. Nelson chooses a student taking a history class or a biology class.

After you answered all the parts of Question 3, read the following:

Generalizing the discussion above, we have the following rule:

Addition Rule

Thus, the probability of the event A or B is equal to the probability of A plus the probability of B minus the probability of A and B.

Note that in terms of the operations union and intersection the event A or B corresponds to A union B and the event A and B corresponds to A intersection B. Thus we can rewrite the addition rule as

4. A card is selected at random from a deck of 52 cards. Use the addition rule to show that the probability the card chosen is a queen or a diamond is 4/13.

5. If a fair die is rolled twice, what is the probability that either the sum of the face is 8 or at least one roll is 5?