6.2 Sets and Probability - A useful application of set theory is in an area of mathematics known as probability. Example 1 To determine which football team will kick off to begin the game, a coin is tossed in the air. How likely is it that the coin will land heads? Probability Definition: Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. P(E) = the number of ways event E may occur the total number of possible outcomes Our set theory notation gives us an easy way, to evaluate the probability fraction for a particular event. We define a sample space to be the set of all possible outcomes that may occur when we perform an experiment The sample space corresponds to the universe in set theory. In Example 1, the experiment is tossing a coin, and the sample space would be the set = {heads, tails} The event whose probability we wish to determine would be the set E = {heads} The outcome "heads" is called a sample point in the sample space. Notice that E must be a subset of S. Since the number of ways event E can occur is the same as the number of elements in set E, and the total number of possible outcomes is the same as the number of elements in S P(E) = the number of elements in set E the number of elements in the entire sample space S P(E) = P(heads) = the number of ways event E may occur the total number of possible outcomes = 1 2 Example 2 John Zakie has complained that when course registration is done in alphabetical order, the people whose last names start with "A" always get first choice. In order to make the registration process fairer, it was decided to hold an alphabet lottery to pick the letter of the last name with which to begin the registration process. Since there are 26 letters in the alphabet, 26 slips of paper, each containing a different letter of the alphabet, are placed in a jar and then one is selected. We assume that each slip of
paper has an equal opportunity of being chosen. How likely is it that the letter "z" will be the first one chosen? How likely is it that the first letter chosen will be a vowel? P(z will be chosen) = 1 To determine how likely the first letter chosen will be a vowel we count the number of elements in this event. There are five vowels in the alphabet. They form the set E = {a,e,i,o,u}. Then, 26 P(E) = P(vowel) = the number of ways event E may occur the total number of possible outcomes = 5 26 Example 3 A die is a small cube with dots on each of its six faces, numbered 1 through 6. Suppose a fair die is rolled. What is the probability that the side facing up is the number 5? What is the probability that it is a number less than 3? What is the probability that it is an even number? Example 4 An urn contains seven colored chips: four green chips, two red chips and one blue chip. One chip is selected at random. Determine: P(green) P(red) P(blue). Mutually Exclusive Events In Example 4, there are only three possible events that can occur if we select a chip at random. The chip must be green, red, or blue. These events are all mutually exclusive, that is, they cannot happen at the same time. P(green) + P(red) + P(blue) = 4 7 + 2 7 + 1 7 = 1
The sum of the probabilities of all the mutually exclusive events in a sample space is always one. Complementary Probability If P(E) represents the probability that event E occurs, we let P(E') represent the probability that event E will not occur. We call P(E') the complementary probability of event E. Example 5 Consider a deck of 52 ordinary playing cards. One card is selected at random. Determine the probability of not getting an ace. Example 6 A die is rolled. Find the probability that the number rolled is not greater than four. Probability and Percents Sometimes we may not know the total number of elements in a sample space, but we may be given the relationship among the various events in terms of percentages. Example 7 At a certain movie theater, 70% of the women are wearing shoes, 20% are wearing sneakers, and 10% are wearing boots. If a woman is chosen at random, find: the probability that she is wearing shoes the probability that she is wearing sneakers the probability that she is not wearing boots.
In-Class Exercises 1. Twenty-six blocks, each containing one letter of the alphabet, are placed in a carton. One of the blocks is picked at random. Find the probability that the block is a. the letter "x". b. the letter "p". c. not the letter "q". d. not a vowel. e. a letter contained in the word "dog". f. not a letter contained in the word "frog". g. neither an "a" nor a "b". 2. A die is rolled. Find: a) P(2) b) P(1) + P(2) + P(3) + P(4) + P(5) + P(6). c) P(2or3) d) P(odd) e) P(not 4) f) P(5') g) P(neither 5 nor 6). h) P(7) i) P(less than 4). j) P(not less than 3). k) P((greater than 5)'). l) P((less than 9)').
3. The results for a television survey at 9 pm on a certain night were as follows: 20% of the homes surveyed were tuned to NBC, 10 % were tuned to ABC, 5% were tuned to CNN and 15% were tuned to FOX. The survey company believes that this reflects the viewing habits of the entire nation. If a home is called at random and one television is on in that home, find the probability that the television is tuned to: a. CNN. b. either NBC or FOX. c. one of the four networks cited in the survey. d. none of the four networks cited in the survey. 4. A card is selected from an ordinary deck of fifty-two cards. Find: a. P(club). b. P(not a club). c. P(not a black card). d. P(not a king). e. P(the six of spades). f. P(a queen or a king). g. P(neither a queen nor a king). h. P(the card is less than five), assuming that the ace counts as a one. 5. An urn contains seven colored chips: two red, four green and one blue. Find the probability that a chip selected at random is a. red. b. green. c. blue. d. not red. e. not blue. f. neither green nor blue. g. white. h. red or green.