Homework 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.

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1 Homework 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

2 Warm Up 1. If you roll a number cube, what are the possible outcomes? 2. Add +. 3. Add +. 1, 2, 3, 4, 5, or 6 1 12 1 6 1 2 2 36 1 4 5 9

3 Vocabulary theoretical probability equally likely fair mutually exclusive disjoint events

4 Theoretical probability is used to estimate probabilities by making certain assumptions about an experiment. Suppose a sample space has 5 outcomes that are equally likely, that is, they all have the same probability, x. The probabilities must add to 1. x + x + x + x + x = 1 5x = 1 x = 1 5

5 A coin, die, or other object is called fair if all outcomes are equally likely.

6 An experiment consists of spinning this spinner once. Find the probability of each event. Additional Example 1: Calculating Theoretical Probability A. 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

7 Additional Example 1: Calculating Theoretical Probability B. 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 = An experiment consists of spinning this spinner once. Find the probability of each event.

8 Check It Out! Example 1 A. P(1) An experiment consists of spinning this spinner once. Find the probability of each event. 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

9 Check It Out! Example 1 B. 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 = An experiment consists of spinning this spinner once. Find the probability of each event.

10 An experiment consists of rolling one fair number cube and flipping a coin. Find the probability of the event. Additional Example 2: Calculating Probability for a Fair Number Cube and a Fair Coin A. 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. 1H2H3H4H5H6H 1T2T3T4T5T6T

11 Additional Example 2: Calculating Theoretical Probability for a Fair Coin B. 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 An experiment consists of rolling one fair number cube and flipping a coin. Find the probability of the event.

12 Check It Out! Example 2 A. 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 2 An experiment consists of flipping two coins. Find the probability of each event.

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

14 Stephany has 2 dimes and 3 nickels. How many pennies should be added so that the probability of drawing a nickel is ? Additional Example 3: Altering Probability Adding pennies to the bag will increase the number of possible outcomes. Set up a proportion. Let x equal the number of pennies 3 5 + x = 3 7 3(5 + x) = 3(7) Find the cross products. 3 7

15 Additional Example 3 Continued Multiply. 15 + 3x = 21 –15 – 15 Subtract 15 from both sides. 3x = 6 x = 2 Divide both sides by 3. 2 pennies should be added to the bag. 3

16 Carl has 3 green buttons and 4 purple buttons. How many white buttons should be added so that the probability of drawing a purple button is ? Check It Out! Example 3 Adding buttons to the bag will increase the number of possible outcomes. Let x equal the number of white buttons. Set up a proportion. Let x equal the number of white buttons. 4 7 + x = 2 9 2(7 + x) = 9(4) Find the cross products. 2 9

17 Check It Out! Example 3 Continued Multiply. 14 + 2x = 36 –14 – 14Subtract 14 from both sides. 2x = 22 x = 11 Divide both sides by 2. 11 white buttons should be added to the bag. 2

18 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.

19 Additional Example 4: Finding the Probability of Mutually Exclusive Events Suppose you are playing a game in which you roll two fair number cubes. 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 the game will end on your next roll? It is impossible to roll both a total of 5 and a total of 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 flip.

20 Additional Example 4 Continued The event “total = 2” consists of 1 outcome, (1, 1), so P(total = 2) =. 1 36 The probability the game will end is, or about 14%. 5 36 P(game ends) = P(total = 2) + P(total = 5) The event “total = 5” consists of 4 outcomes, (1, 4), (2, 3), (4, 1), (3, 2), so P(total = 5) =. 4 36 5 = + = 1 36 4

21 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.

22 Check It Out! Example 4 Continued P(game ends) = P(both tails) + P(both heads) 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 1 4 = + 1 4 1 4 = 1 2 The probability that the game will end is, or 50%. 1 2

23 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