Unit 1: Outcomes & Liklihoods

1. Unit 1:Outcomes & Liklihoods

2. Probability: THEORETICAL EXPERIMENTAL

3. An event is any set of one or more outcomes. The probability of an event, written P(event), is a number from 0 (or 0%) to 1 (or 100%) that tells you how likely the event is to happen. • A probability of 0 means the event is impossible, or can never happen. • A probability of 1 means the event is certain, • or has to happen. • The probabilities of all the outcomes in the • sample space add up to 1.

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

5. Two events are mutually exclusive, ordisjoint 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. Suppose both A and B are two mutually exclusive events. • P(both A and B will occur) = 0 • P(either AorB will occur) = P(A) + P(B)

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

7. 1 5 Example 1: 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. P(4) = = number of outcomes for 4 5

8. Example 2: 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 In other words… A coin has 2 sides, a cube has 6 sides: 2 x 6 = 12 12 possible outcomes in the sample space.

9. 3 3 7 7 3 = 5 + x Example 3: Stephany has 2 dimes and 3 nickels. How many pennies should be added so that the probability of drawing a nickel is ? Adding pennies to the bag will increase the number of possible outcomes. Let x equal the number of pennies. Set up a proportion. Use cross multiplication. 3(5 + x) = 3(7)

10. 3 3 Example 3 Continued: 3(5 + x) = 3(7) 15 + 3x = 21 Multiply. Subtract 15 from both sides. –15 – 15 3x = 6 Divide both sides by 3. x = 2 2 pennies should be added to the bag.

11. The event “total = 2” consitsof 1 outcome, (1, 1), so P(total = 2) = . 1 1 1 36 36 36 = The probability that you will lose is , or about 3%. Example 4: 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 you will lose on your next roll? (Cubes = 6 sides…so… 6 x 6 = 36) P(game ends) = P(total = 2)

12. In experimental probability, the likelihood of an event is estimated by repeating an experiment many times and observing the number of times the event happens. That number is divided by the total number of trials. The more the experiment is repeated, the more accurate the estimate is likely to be. number of times the event occurs total number of trials probability 

13. number of red marbles drawn 1550 = total number of marbles drawn Example 1: A marble is randomly drawn out of a bag and then replaced. The table shows the results after 50 draws. Estimate the probability of drawing a red marble. probability  The probability of drawing a red marble is about 0.3, or 30%.

14. Example 2: Use the table to compare the probability that the Huskies will win their next game with the probability that the Knights will win their next game.

15. number of wins probability  total number of games 79 probability for a Huskies win   0.572 138 90 probability for a Knights win   0.616 146 Example 2 Continued The Knights are more likely to win their next game than the Huskies.