1. Chapter 22 Probability

2. An experiment is an activity involving chance. Each repetition or observation of an experiment is a trial, and each possible result is an outcome. The sample space of an experiment is the set of all possible outcomes.

3. Example 1: Identifying Sample Spaces and Outcomes Identify the sample space and the outcome shown for each experiment. A. Rolling a die Sample space:{1, 2, 3, 4, 5, 6} Outcome shown: 4 B. Spinning a spinner Sample space: {red, green, orange, purple} Outcome shown: green

4. An event is an outcome or set of outcomes in an experiment. Probability is the measure of how likely an event is to occur. Probabilities are written as fractions or decimals from 0 to 1, or as percents from 0% to 100%.

5. Impossible As likely as not Certain Unlikely Likely 0% Events with a probability of 0% never happen. 50% Events with a probability of 50% have the same chance of happening as not. 100% Events with a probability of 100% always happen.

6. Example 2: Estimating the Likelihood of an Event State impossible, unlikely, as likely as not, likely, or certain to describe each event. A. A shoe selected from a pair of shoes fits the right foot. as likely as not B. Wentz dating Scarlett Johansson.Very Likely C. A pack of wild monkeys throwing bananas at Annika. Certain D. A randomly selected month contains the letter R. likely

7. You can estimate the probability of an event by performing an experiment. The experimental probability of an event is the ratio of the number of times the event occurs to the number of trials. The more trials performed, the more accurate the estimate will be.

8. Relative Frequency This gives information about how often an event occurred compared with other events. eg: Maths Exam results from 26 students Exam %No. of students Relative Frequency > 8010 60 - 8012 40 - 603 < 401 = 0.385 = 0.462 = 0.115 = 0.0385

9. Example 3A: Finding Experimental Probability OutcomeFrequency Green15 Orange10 Purple8 Pink7 An experiment consists of spinning a spinner. Use the results in the table to find the experimental probability of the event. Spinner lands on orange

10. Example 3B: Finding Experimental Probability OutcomeFrequency Green15 Orange10 Purple8 Pink7 An experiment consists of spinning a spinner. Use the results in the table to find the experimental probability of the event. Spinner does not land on green

11. Check It Out! Example 3a An experiment consists of spinning a spinner. Use the results in the table to find the experimental probability of each event. Spinner lands on red OutcomeFrequency Red7 Blue8 Green5

12. Check It Out! Example 3b Spinner does not land on red An experiment consists of spinning a spinner. Use the results in the table to find the experimental probability of each event. OutcomeFrequency Red7 Blue8 Green5

13. You can use experimental probability to make predictions. A prediction is an estimate or guess about something that has not yet happened.

14. Example 4A: Quality Control Application A manufacturer inspects 500 strollers and finds that 498 have no defects. What is the experimental probability that a stroller chosen at random has no defects? Find the experimental probability that a stroller has no defects. = 99.6% The experimental probability that a stroller has no defects is 99.6%.

15. Example 4B: Manufacturing Application A manufacturer inspects 500 strollers and finds that 498 have no defects. The manufacturer shipped 3500 strollers to a distribution center. Predict the number of strollers that are likely to have no defects. Find 99.6% of 3500. 0.996(3500) = 3486 The prediction is that 3486 strollers will have no defects.

16. Check It Out! Example 4a A manufacturer inspects 1500 electric toothbrush motors and finds 1497 have no defects. What is the experimental probability that a motor chosen at random will have no defects? Find the experimental probability that a motor has no defects. = 99.8%

17. Check It Out! Example 4b A manufacturer inspects 1500 electric toothbrush motors and finds 1497 have no defects. There are 35,000 motors in a warehouse. Predict the number of motors that are likely to have no defects. Find 99.8% of 35,000. 0.998(35000) = 34930 The prediction is that 34,930 motors will have no defects.

18. Homework Page 570 1 - 4

19. Chapter 22 B Sample Space

20. Grid Diagrams A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. Graph the sample space and use it to give the probabilities: (a) P(head and a 4)(b) P(head or a 4) Sample space P(head and a 4) = P(head or a 4) = (those shaded)

21. Sample Space is {H1, H2, H3, H4, T1, T2, T3, T4} A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. We can work out the sample space by a grid diagram Grid Diagram

22. Grid Diagrams The sample space when rolling 2 dice can be shown by the following lattice diagram: 1 2 3 4 5 6 123456123456 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Pr (double) == Pr (total ≥ 7) = =

23. Using Grids to find Probabilities red die blue die 621 1 2 3 4 5 6 5 43 ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● P(double) = = P(total ≥ 7) = = coin die 6 5 4321 H T ● ● ● ● ● ●● ● ● ● ● ● Rolling 2 dice: Rolling a die & Flipping a coin: P(tail and a 5) = P(tail or a 5) =

24. Tree Diagrams

25. Tree Diagrams for Probability Tree diagrams are useful to work out probabilities. B B B B B B B G G G G G G G eg: Show possible combination of genders in a 3 child family 1 st child 2 nd child 3 rd child Outcomes BBB BBG BGB BGG GBB GBG GGB GGG P(2 girls & a boy) =   

26. For example – 10 colored beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, color noted, returned to bag, then a second taken. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG INDEPENDENT EVENTS

27. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG 0.3 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 Probabilities P(RR) = 0.3x0.3 = 0.09 P(RB) = 0.3x0.2 = 0.06 P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 P(BB) = 0.2x0.2 = 0.04 P(BG) = 0.2x0.5 = 0.10 P(GR) = 0.5x0.3 = 0.15 P(GB) = 0.5x0.2 = 0.10 P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0

28. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG 0.3 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 Probabilities P(RR) = 0.3x0.3 = 0.09 P(RB) = 0.3x0.2 = 0.06 P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 P(BB) = 0.2x0.2 = 0.04 P(BG) = 0.2x0.5 = 0.10 P(GR) = 0.5x0.3 = 0.15 P(GB) = 0.5x0.2 = 0.10 P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0

29. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG 0.3 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 Probabilities P(RR) = 0.3x0.3 = 0.09 P(RB) = 0.3x0.2 = 0.06 P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 P(BB) = 0.2x0.2 = 0.04 P(BG) = 0.2x0.5 = 0.10 P(GR) = 0.5x0.3 = 0.15 P(GB) = 0.5x0.2 = 0.10 P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0

30. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG 0.3 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 Probabilities P(RR) = (0.3)(0.3) = 0.09 P(RB) = (0.3)(0.2) = 0.06 P(RG) = (0.3)(0.5) = 0.15 P(BR) = (0.2)(0.3) = 0.06 P(BB) = (0.2)(0.2) = 0.04 P(BG) = (0.2)(0.5) = 0.10 P(GR) = (0.5)(0.3) = 0.15 P(GB) = (0.5)(0.2) = 0.10 P(GG) = (0.5)(0.5) = 0.25 All ADD UP to 1.0

31. Homework Page 575 1 - 3