12.3 An Introduction to Probability

1. 12.3 An Introduction to Probability What is probability? What is the difference between theoretical probability and experimental probability? What is geometric probability?

2. THEORETICAL AND EXPERIMENTAL PROBABILITY The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur. There are two types of probability: theoretical and experimental.

3. THEORETICAL AND EXPERIMENTAL PROBABILITY THE THEORETICAL PROBABILITY OF AN EVENT 4 P (A) = 9 total number of outcomes all possible outcomes You can express a probability as a fraction, a decimal, or a percent.For example: , 0.5, or 50%. 1 2 The theoretical probability of an event is often simply called the probability of the event. When all outcomes are equally likely, the theoretical probability that an event Awill occur is: number of outcomes in A P (A) = outcomes in event A

4. Finding Probabilities of Events 1 = 6 number of ways to roll the die You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling a 4. SOLUTION Only one outcome corresponds to rolling a 4. number of ways to roll a 4 P (rolling a 4) =

5. Finding Probabilities of Events 3 1 = = 2 6 number of ways to roll the die You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling an odd number. SOLUTION Three outcomes correspond to rolling an odd number: rolling a 1, 3, or a 5. number of ways to roll an odd number P (rolling odd number) =

6. Finding Probabilities of Events 6 = = 1 number of ways to roll the die 6 You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling a number less than 7. SOLUTION All six outcomes correspond to rolling a number less than 7. number of ways to roll less than 7 P (rolling less than 7 ) =

7. Probabilities Involving Permutations or Combinations 1 1 P(playing 8 in order) = =  0.0000248 8! 40, 320 You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song. What is the probability that the songs are played in the same order they are listed on the CD? Help SOLUTION There are 8! differentpermutations of the 8 songs. Of these, only 1 is the order in which the songs are listed on the CD. So, the probability is:

8. Probabilities Involving Permutations or Combinations 4 C 2 6 3 P(playing 2 favorites first) = = =  0.214 8 C 2 28 14 You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song. You have 4 favorite songs on the CD. What is the probability that 2 of your favorite songs are played first, in any order? Help SOLUTION There are 8C2 different combinations of 2 songs. Of these, 4C2 contain 2 of your favorite songs. So, the probability is:

9. Probabilities Involving Permutations or Combinations Sometimes it is not possible or convenient to find thetheoretical probability of an event. In such cases youmay be able to calculate an experimental probabilityby performing an experiment, conducting a survey, orlooking at the history of the event.

10. Finding Experimental Probabilities In 1998 a survey asked Internet users for their ages. The results are shown in the bar graph.

11. Finding Experimental Probabilities 1636 6617 3693 491 6 1636 P(user is at most 20) =  0.131 12,443 Find the experimental probability that a randomly selected Internet user is at most 20 years old. SOLUTION The number of people surveyed was 1636 + 6617+ 3693 + 491 + 6 = 12,443. Of the people surveyed, 1636 are at most 20 years old. So, the probability is:

12. Finding Experimental Probabilities 4190 P(user is at least 41) =  0.337 12,443 Find the experimental probability that a randomly selected Internet user is at least 41 years old. Given that 12,443 people were surveyed. SOLUTION Of the people surveyed, 3693 + 491 + 6 = 4190 are at least 41 years old. So, the probability is:

13. GEOMETRIC PROBABILITY Some probabilities are found by calculating a ratio oftwo lengths, areas, or volumes. Such probabilities arecalled geometric probabilities.

14. Using Area to Find Probability You throw a dart at the board shown. Your dart is equallylikely to hit any point inside the square board. Are you more likely to get 10 points or 0 points?

15. Using Area to Find Probability area of entire board  9  • 32 = = =  0.0873 182 324 36 area of entire board 324 – 81 4 –  182 – (• 9 2 ) = = =  0.215 182 324 4 Are you more likely to get 10 points or 0 points? SOLUTION area of smallest circle P (10 points) = area outside largest circle P (0 points) = You are more likely to get 0 points.

16. What is probability? A number between 0 and 1 of the likelihood that an event will occur. • What is the difference between theoretical probability and experimental probability? When all outcomes are equally likely (theoretical) compared to the actual information collected by performing an experiment, conducting a survey or looking at the history of an event (experimental). • What is geometric probability? Probabilities using the ratio of two lengths, areas or volumes.

17. Assignment: P. 719 12-29, 41- 43