Learn to estimate probability using theoretical methods.

Vocabulary equally likely theoretical probability fair geometric probability mutually exclusive disjoint events

When the outcomes in a sample space have an equal chance of occurring, the outcomes are said to be equally likely. The theoretical probability of an event is the ratio of the number of ways the event can occur to the total number of equally likely outcomes. A coin, number cube, or other object is called fair if all outcomes are equally likely.

Additional Example 1A: Calculating Theoretical Probability An experiment consists of spinning this spinner once. Find the probability of each event. P(4) The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and 5. 1 5 P(4) = = number of outcomes for 4 5

Additional Example 1B: Calculating Theoretical Probability An experiment consists of spinning this spinner once. Find the probability of each event. P(even number) There are 2 outcomes in the event of spinning an even number: 2 and 4. P(even number) = number of possible even numbers 5 2 5 =

Check It Out: Example 1A An experiment consists of spinning this spinner once. Find the probability of each event. P(1) The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and 5. 1 5 P(1) = = number of outcomes for 1 5

Check It Out: Example 1B An experiment consists of spinning this spinner once. Find the probability of each event. P(odd number) There are 3 outcomes in the event of spinning an odd number: 1, 3, and 5. P(odd number) = number of possible odd numbers 5 3 5 =

Additional Example 2A: Calculating Probability for a Fair Number Cube and a Fair Coin An experiment consists of rolling one fair number cube and flipping a coin. Find the probability of the event. Show a sample space that has all outcomes equally likely. The outcome of rolling a 5 and flipping heads can be written as the ordered pair (5, H). There are 12 possible outcomes in the sample space. 1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T

Additional Example 2B: Calculating Theoretical Probability for a Fair Coin An experiment consists of flipping a coin. Find the probability of the event. P(tails) There are 6 outcomes in the event “flipping tails”: (1, T), (2, T), (3, T), (4, T), (5, T), and (6, T). P(tails) = = 6 12 1 2

Check It Out: Example 2A An experiment consists of flipping two coins. Find the probability of each event. P(one head & one tail) There are 2 outcomes in the event “getting one head and getting one tail”: (H, T) and (T, H). P(head and tail) = = 2 4 1

Check It Out: Example 2B An experiment consists of flipping two coins. Find the probability of each event. P(both tails) There is 1 outcome in the event “both tails”: (T, T). P(both tails) = 1 4

Theoretical probability that is based on the ratios of geometric lengths, areas, or volumes is called geometric probability.

Additional Example 3: Finding Geometric Probability Find the probability that a point chosen randomly inside the circle is within the shaded region. Round to the nearest hundredth. probability = area of circle – area of triangle area of circle The probability that a point chosen within the circle is within the shaded region is

Check It Out: Example 3 Find the probability that a point chosen randomly inside the circle is within triangle. Round to the nearest hundredth. probability = area of triangle area of circle The probability that a point chosen within the circle is within the shaded region is

Two events are mutually exclusive, or disjoint events, if they cannot both occur in the same trial of an experiment. For example, rolling a 5 and an even number on a number cube are mutually exclusive events because they cannot both happen at the same time.

Additional Example 4: Find the Probability of Mutually Exclusive Events Suppose you are playing a game in which you roll two fair dice. If you roll a total of five you will win. If you roll a total of two, you will lose. If you roll anything else, the game continues. What is the probability that you will lose on your next roll? It is impossible to roll a total of 5 and a total 2 at the same time, so the events are mutually exclusive. Add the probabilities to find the probability of the game ending on your next roll.

Additional Example 4 Continued 36 The event “total = 5” consists of 4 outcomes, (1, 4), (2, 3), (3, 2), and (4, 1). So P(total = 5) = . The event “total = 2” consists of 1 outcome, (1, 1), so P(total = 2) = . 1 36 P(game ends) = P(total = 5) + P(total = 2) 4 36 = 1 + The probability that the game will end is , or about 13.9%. 5 36

Check It Out: Example 4 Suppose you are playing a game in which you flip two coins. If you flip both heads you win and if you flip both tails you lose. If you flip anything else, the game continues. What is the probability that the game will end on your next flip? It is impossible to flip both heads and tails at the same time, so the events are mutually exclusive. Add the probabilities to find the probability of the game ending on your next flip.

Check It Out: Example 4 Continued The event “both heads” consists of 1 outcome, (H, H), so P(both heads) = . The event “both tails” consists of 1 outcome, (T, T), so P(both tails) = . 1 4 P(game ends) = P(both tails) + P(both heads) = + 1 4 = 1 2 The probability that the game will end is , or 50%. 1 2

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 20

Lesson Quiz An experiment consists of rolling a fair number cube. Find each probability. 1. P(rolling an odd number) 2. P(rolling a prime number) An experiment consists of rolling two fair number cubes. Find each probability. 3. P(rolling two 3’s) 4. P(total shown > 10) 1 2 1 2 1 36 1 12

Lesson Quiz for Student Response Systems 1. An experiment consists of spinning this spinner once. Identify P(odd number). A. B. 2 C. D. 3 25 35 22

Lesson Quiz for Student Response Systems 2. An experiment consists of spinning this spinner once. Identify P(not 8). A. B. C. D. 12 13 23 78 23

Lesson Quiz for Student Response Systems 3. An experiment consists of tossing two fair coins at the same time. Identify P(at least one head). A. B. C. D. 1 12 14 34 24