1. 1 Lesson 6.2.4 Probability of an Event Not Happening Probability of an Event Not Happening

2. 2 Lesson 6.2.4 Probability of an Event Not Happening California Standard: Statistics, Data Analysis and Probability 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1– P is the probability of an event not occurring. What it means for you: You’ll learn how to find the probability that a particular event does not happen. Key words: probability event outcome favorable

3. 3 Probability of an Event Not Happening Lesson 6.2.4 The probabilities you’ve seen so far represent the chances that an event will happen. In this Lesson, you’ll learn how to find the probability that an event doesn’t happen. You’ll be pleased to know that the math you’ll need to do isn’t much different from what you’ve been doing so far in this Section.

4. 4 Probability of an Event Not Happening Find Probabilities by Counting Outcomes Lesson 6.2.4 So far you have worked out probabilities of events happening. You can also find the probability that an event doesn’t happen by counting the number of outcomes that don’t match the event. P(red) = 0.25

5. 5 Probability of an Event Not Happening Example 1 Lesson 6.2.4 Solution Solution follows… Ruben spins this spinner. Find the probability that Ruben spins the color yellow. What is the probability that Ruben does not spin the color yellow? There are 3 possible outcomes: red, yellow, and blue. So P(Yellow) = 1 3 If he doesn’t spin yellow, Ruben must spin either red or blue. So there are 2 favorable outcomes for not spinning yellow. So P(Not yellow) = P(Red or blue) = 2 3

6. 6 Probability of an Event Not Happening Guided Practice Solution follows… Lesson 6.2.4 In Exercises 1–8, determine each probability for one spin of the spinner on the right. Give your answers as simplified fractions. 1. P(Green)2. P(Not green) 3. P(Yellow)4. P(Not yellow) 5. P(Red)6. P(Not red) 7. P(Orange)8. P(Not orange) 1 3 2 6 = 2 3 4 6 = 1 6 5 6 1 3 2 6 = 2 3 4 6 = 01

7. 7 Probability of an Event Not Happening You Can Find P(not A) by Counting Outcomes Lesson 6.2.4 You can use the probability that event A will happen to find the probability that A won’t happen. Example 2 shows you how.

8. 8 Probability of an Event Not Happening Example 2 Lesson 6.2.4 Solution follows… Mario is playing a game using this spinner. He spins it twice, and adds the numbers he spins to get his score. This table gives the possible outcomes. If A is the event “Mario scores 8,” then find: (i) P(A), (ii) P(not A). What do you notice about P(A) + P(not A)? 2468 46810 68 12 8101514 1357 1st spin + 1 3 5 7 2nd spin 13 57

9. 9 Solution continues… Probability of an Event Not Happening Example 2 Lesson 6.2.4 Solution If A is the event “Mario scores 8,” then find: (i) P(A), (ii) P(not A). 2468 46810 68 12 8101514 1357 1st spin + 1 3 5 7 2nd spin (i) The table shows there are 16 possible outcomes. There are 4 possible outcomes where Mario does score 8. So P(Mario scores 8) = = (or 25%). 4 16 1 4

10. 10 Probability of an Event Not Happening Example 2 Lesson 6.2.4 Solution (continued) If A is the event “Mario scores 8,” then find: (i) P(A), (ii) P(not A). 2468 46810 68 12 8101514 1357 1st spin + 1 3 5 7 2nd spin So P(Mario does not score 8) = = (or 75%). 12 16 3 4 Solution continues… (ii) But if there are 4 possible outcomes where Mario does score 8, then there must be 16 – 4 = 12 outcomes where Mario does not score 8.

11. 11 Probability of an Event Not Happening Example 2 Lesson 6.2.4 What do you notice about P(A) + P(not A)? 3 4 P(A) = (or 25%) and P(not A) = (or 75%) 1 4 So, P(A) + P(not A) = + = 1 (or 100%). 1 4 3 4 Solution (continued) If A is the event “Mario scores 8,” then find: (i) P(A), (ii) P(not A).

12. 12 Probability of an Event Not Happening You Can Also Find P(not A) If You Know P(A) Lesson 6.2.4 The result of P(A) and P(not A) adding up to 1 is always true. P(event A happening) + P(event A not happening) = 1 = 100% You can write this rule as P(A)+ P(not A) = 1.

13. 13 Probability of an Event Not Happening Example 3 Lesson 6.2.4 Solution Solution follows… The weather channel says there is 30% chance of rain today. What is the probability that it will not rain today? P(rain) + P(not rain) = 100% So 30% + P(not rain) = 100% This means P(not rain) = 100% – 30% = 70%

14. 14 Probability of an Event Not Happening Guided Practice Solution follows… Lesson 6.2.4 Exercises 9–16 give P(A). Find P(not A) in each case. 9. P(A) = 10. P(A) = 0.58 11. P(A) = 11% 12. P(A) = 43% 13. P(A) = 0.9 14. P(A) = 15. P(A) = 1 16. P(A) = 0 3 4 11 12 3 4 1 4 1 – = 11 12 1 – = 1 12 1 – 0.58 = 0.42 100% – 43% = 57% 1 – 0 = 1 100% – 11% = 89% 1 – 1 = 0 1 – 0.9 = 0.1

15. 15 Probability of an Event Not Happening Guided Practice Solution follows… Lesson 6.2.4 Find the following probabilities for Mario’s game from Example 2. The table below gives the possible outcomes. Write your answers as decimals 17. P(Mario scores 14) 18. P(Mario doesn’t score 14) 19. P(Mario scores 6) 20. P(Mario doesn’t score 6) 2468 46810 68 12 8101514 1357 1st spin + 1 3 5 7 2nd spin 1 ÷ 16 = 0.0625 1 – 0.0625 = 0.9375 3 ÷ 16 = 0.1875 1 – 0.1875 = 0.8125

16. 16 Probability of an Event Not Happening Guided Practice Solution follows… Lesson 6.2.4 Find the following probabilities for Mario’s game from Example 2. The table below gives the possible outcomes. Write your answers as decimals 21. P(Mario scores less than 10) 22. P(Mario doesn’t score less than 10) 23. P(Mario scores 7) 24. P(Mario doesn’t score 7) 2468 46810 68 12 8101514 1357 1st spin + 1 3 5 7 2nd spin 10 ÷ 16 = 0.625 1 – 0.625 = 0.375 0 ÷ 16 = 0 1 – 0 = 1

17. 17 Probability of an Event Not Happening Guided Practice Solution follows… Lesson 6.2.4 25. Jenna rolls a die. Event A is “rolling a number greater than 4.” Event B is “rolling a number less than 4.” Jenna claims that P(event A) + P(event B) = 1. Explain to Jenna why this is not true. For the statement P(event A) + P(event B) = 1 to be true, event B must be the same as event A not happening. The favorable outcomes for event A are rolling a 5 or a 6, so for event A to not happen, you must roll a 1, 2, 3, or 4. But, the favorable outcomes for event B are rolling a 1, 2, or 3, which is not the same as event A not happening, so the statement is false.

18. 18 Probability of an Event Not Happening Independent Practice Solution follows… Lesson 6.2.4 1. The probability of an event occurring is 0.6. Explain why the probability of the event not occurring cannot be 0.2. 2. P(A occurring) and P(A not occurring) are the same. Find P(A). 3. P(A occurring) is twice P(A not occurring). Find P(A). The probabilities of an event occurring and not occurring must add up to 1. 0.6 + 0.2 = 0.8, and not 1. P(A) + P(not A) = 1, and P(A) = P(not A)  P(A) = 1 2 P(A) + P(not A) = 1, and P(A) = 2 × P(not A)  P(A) = 2 3

19. 19 Calculate P(A) and P(not A), where event A is Cynthia picking: 4. At least one heart5. Two diamonds 6. At least one red card7. One heart and one club 8. Two cards the same suit Probability of an Event Not Happening Independent Practice Solution follows… Lesson 6.2.4 Cynthia picks one card from a standard pack of 52 cards. She makes a note of the suit, then replaces the card and picks another one. The tree diagram shows the possible outcomes. Calculate P(A) and P(not A), where event A is Cynthia picking: 4. At least one heart5. Two diamonds 6. At least one red card7. One heart and one club 8. Two cards the same suit D = diamonds, S = spades, C = clubs, H = hearts, DD DC DH DS CD CC CH CS HD HC HH HS SD SC SH SS Card 1 Card 2 Outcomes 7 16 P(A) =, P(not A) = 9 16 1 P(A) =, P(not A) = 15 16 3 4 1 4 P(A) =, P(not A) = 1 8 7 8 1 4 3 4 D = diamonds, S = spades, C = clubs, H = hearts, DD DC DH DS CD CC CH CS HD HC HH HS SD SC SH SS Card 1 Card 2 Outcomes

20. 20 Probability of an Event Not Happening Lesson 6.2.4 Round Up The rule P(A) + P(not A) = 1 is very useful and important. If you need to find a complicated-looking P(not A), the first thing you should think is “Is it easier to find 1 – P(A)?” The answer is quite often yes.