Section 12.2 Theoretical Probability

What You Will Learn Equally Likely Outcomes Theoretical Probability

Equally Likely Outcomes If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes. For equally likely outcomes, the probability of Event E may be calculated with the following formula.

Example 1: Determining Probabilities A die is rolled. Find the probability of rolling a) a 5. b) an even number. c) a number greater than 3. d) a 7. e) a number less than 7.

Example 1: Determining Probabilities Solution a) b) Rolling an even number can occur in three ways: 2, 4 or 6.

Example 1: Determining Probabilities Solution c) Three numbers are greater than 3: 4, 5 or 6.

Example 1: Determining Probabilities Solution d) No outcomes will result in a 7. Thus, the event cannot occur and the probability is 0.

Example 1: Determining Probabilities Solution e) All the outcomes 1 through 6 are less than 7. Thus, the event must occur and the probability is 1.

Important Probability Facts The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1 inclusive; that is, 0 ≤ P(E) ≤ 1. The sum of the probabilities of all possible outcomes of an experiment is 1.

The Sum of the Probabilities Equals 1 P(A) + P(not A) = 1 or P(not A) = 1 – P(A)

Example 3: Selecting One Card from a Deck A standard deck of 52 playing cards is shown.

Example 3: Selecting One Card from a Deck The deck consists of four suits: hearts, clubs, diamonds, and spades. Each suit has 13 cards, including numbered cards ace (1) through 10 and three picture (or face) cards, the jack, the queen, and the king.

Example 3: Selecting One Card from a Deck Hearts and diamonds are red cards; clubs and spades are black cards. There are 12 picture cards, consisting of 4 jacks, 4 queens, and 4 kings. One card is to be selected at random from the deck of cards. Determine the probability that the card selected is

Example 3: Selecting One Card from a Deck a) a 7. b) not a 7. c) a diamond. d) a jack or queen or king (a picture card). e) a heart and spade. f) a card greater than 6 and less than 9.

Example 3: Selecting One Card from a Deck Solution a) a 7. There are 4 7’s in a deck of cards. b) not a 7.

Example 3: Selecting One Card from a Deck Solution c) a diamond. There are 13 diamonds in the deck.

Example 3: Selecting One Card from a Deck Solution d) a jack or queen or king (a picture card). There are 4 jacks, 4 queens, and 4 kings or a total of 12 picture cards.

Example 3: Selecting One Card from a Deck Solution e) a heart and spade. The word and means both events must occur. This is not possible, that one card is both, the probability = 0.

Example 3: Selecting One Card from a Deck Solution f) a card greater than 6 and less than 9. The cards that are both greater than 6 and less than 9 are 7’s and 8’s. There are 4 7’s and 4 8’s, or 8 total.