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1 Lesson 6.2.2 Expressing Probability

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2 Lesson 6.2.2 Expressing Probability 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 meet and use an important formula that will help you to work out probabilities. Key words: probability event outcome favorable

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3 Expressing Probability Lesson 6.2.2 In this Lesson, you’ll learn how to find the probability of an event by looking at all the possible outcomes. You’ll meet a handy formula you can use to do this.

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4 Expressing Probability You Can Calculate Probabilities by Counting Outcomes Lesson 6.2.2 There’s some special probability notation you’ll need. “P(A)” means “the probability that event A will happen.” So for tossing a coin, you can write: “the probability of getting heads is ” as “P(Heads) =.” 1 2 1 2

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5 Expressing Probability Lesson 6.2.2 You can often find probabilities by thinking about the possible outcomes of the situation. Favorable outcomes are outcomes that match the event. P(event) = Number of favorable outcomes Number of possible outcomes

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6 Expressing Probability Example 1 Lesson 6.2.2 Solution Solution follows… Find the probability of rolling a 2 on a standard 6-sided die. The possible outcomes are: 1 2 3 4 5 6 The favorable outcomes are: 2 There are 6 possible outcomes, and 1 favorable outcome. So P(2) = 1 6

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7 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Write the answers to Exercises 1–3 as fractions. Jaden has a bag containing 100 marbles. There are 50 red marbles, 30 blue marbles, and 20 green marbles. He picks out one marble. Find the probability that the marble is: 1. Red 2. Blue 3. Green 1 2 50 out of the 100 marbles are red, so P(red) = = 50 100 3 10 30 out of the 100 marbles are blue, so P(blue) = = 30 100 1 5 20 out of the 100 marbles are green, so P(green) = = 20 100

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8 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Write the answers to Exercises 4–7 as fractions. There are 50 socks in Jasmine’s drawer. There are 25 black socks, 10 blue socks, 10 orange socks, and 5 striped socks. Jasmine picks a sock without looking. Find the probability that the sock is: 4. Black 5. Blue 6. Orange 7. Striped 1 2 25 out of the 50 socks are black, so P(black) = = 25 50 10 out of the 50 socks are blue, so P(blue) = = 1 5 10 50 10 out of the 50 socks are orange, so P(orange) = = 1 5 10 50 5 out of the 50 socks are striped, so P(striped) = = 1 10 5 50

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9 Expressing Probability Example 2 Lesson 6.2.2 Solution Solution follows… What is the probability of picking a queen out of a standard pack of 52 playing cards? Always read the situation carefully. And then think very carefully too. There are 52 possible outcomes. The pack of cards contains 4 queens. So P(queen) = = 4 52 1 13

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10 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Find the probability of the following events when rolling a die once. Give your answers as fractions in their simplest form. 8. Rolling an odd number 9. Rolling an even number 10. Rolling a number more than 2 11. Rolling a prime number There are 3 favorable outcomes (1, 3, 5) out of 6 possible outcomes. 1 2 3 6 P(odd) = = There are 3 favorable outcomes (2, 4, 6) out of 6 possible outcomes. 1 2 3 6 P(even) = = There are 4 favorable outcomes (3, 4, 5, 6) out of 6 possible outcomes. 2 3 4 6 P(>2) = = There are 3 favorable outcomes (2, 3, 5) out of 6 possible outcomes. 1 2 3 6 P(prime) = =

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11 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Find the probability of the following events when picking one card from a standard pack of 52. 12. Picking the jack of diamonds 13. Picking a seven 14. Picking a club 15. Picking a black card There is 1 favorable outcome out of 52 possible outcomes. 1 52 P(JD) = There are 4 favorable outcomes (7D, 7C, 7H, 7S) out of 52 possible outcomes. 4 52 P(7) = = 1 13 There are 13 favorable outcomes (AC – KC) out of 52 possible outcomes. 13 52 P(C) = = 1 4 There are 26 favorable outcomes out of 52 possible outcomes. 26 52 P(black) = = 1 2

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12 Expressing Probability Probability Can Give You Information About Outcomes Lesson 6.2.2 If you know the probability of an event, you might be able to figure out numbers of outcomes. P(event) = Number of favorable outcomes Number of possible outcomes To do this, you need to rearrange this formula:

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13 Expressing Probability Example 3 Lesson 6.2.2 Solution Solution follows… The probability of picking a red marble out of a bag is 25%. There are 20 marbles in the bag. How many red marbles are there? So there are 5 red marbles. P(red) = 25% = = Number of favorable outcomes Number of possible outcomes 1 4 = Number of favorable outcomes 20 1 4 Number of favorable outcomes = 20 × = 5 1 4 Multiply both sides by 20 There are 20 marbles in total

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14 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Exercises 16–17 are about picking one marble (without looking) from a bag full of marbles. Each exercise is about a different bag. 16. P(picking yellow) =, and there are 11 yellow marbles. How many marbles are there altogether? 17. If there are 100 marbles, and P(picking red) = 10%, how many red marbles are there? 1 2 Number of marbles = 11 × 2 = 22 11 Number of marbles = 1 2 Number of red marbles = 100 ÷ 10 = 10 Number of red marbles 100 10% = = 1 10

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15 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Exercises 18–19 are about picking one marble (without looking) from a bag full of marbles. Each exercise is about a different bag. 18. P(picking blue) = 25%, and there are 8 blue marbles. How many marbles are there in total? 19. P(picking black) =. If there are 10 black marbles, how many marbles are there in total? 1 3 Number of marbles = 8 × 4 = 32 8 Number of marbles 25% = = 1 4 Number of marbles = 10 × 3 = 30 10 Number of marbles = 1 3

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16 Expressing Probability Guided Practice Solution follows… Lesson 6.2.2 Exercises 20–21 are about picking one marble (without looking) from a bag full of marbles. Each exercise is about a different bag. 20. There are 80 marbles in the bag in total. P(picking silver) =. How many silver marbles are there? 21. There are 5 green marbles in the bag. What is the total number of marbles if P(picking green) = 20%? 3 4 Number of marbles = 5 × 5 = 25 5 Number of marbles 20% = = 1 5 Number of silver marbles = (3 × 80) ÷ 4 = 60 Number of silver marbles 80 = 3 4

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17 Expressing Probability Independent Practice Solution follows… Lesson 6.2.2 1. The winning contestant on a game show picks one of 80 boxes to find out what prize they win. 1 box contains $1,000,000 4 boxes contain vacation tickets 10 boxes contain cameras 15 boxes contain $10 20 boxes contain a pair of socks 30 boxes contain signed photos of the game show host Find the probability of winning each prize as a fraction. 1 80 3 8 30 80 = 1 4 20 80 = 1 8 10 80 = 4 1 20 = 15 80 = 3 16

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18 Expressing Probability Independent Practice Solution follows… Lesson 6.2.2 Sixty-four raffle tickets have been sold. Maria bought 5 of the tickets and her friend Kyle bought 10. Write the probabilities of the following events in fraction form: 2. Maria will win the raffle. 3. Kyle will win the raffle. 4. Someone other than Maria or Kyle will win the raffle. 5. If Maria wanted to have a 25% chance of winning, how many of the 64 tickets would she need to purchase? 5 64 10 64 5 32 = 49 64 16

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19 Expressing Probability Independent Practice Solution follows… Lesson 6.2.2 Destiny buys 20 tickets in the same raffle as Maria and Kyle, so that 84 tickets have now been sold. Remember, Maria bought 5 tickets and Kyle bought 10. 6. How does this affect Kyle's and Maria's chances of winning? Use percents rounded to the nearest whole percent to compare the probability of winning before and after Destiny's purchase. Kyle’s chance of winning was (10 ÷ 64) × 100 = 16%. It is now (10 ÷ 84) × 100 = 12%. Maria’s chance of winning was (5 ÷ 64) × 100 = 8% It is now (5 ÷ 84) × 100 = 6%. Both their chances of winning have decreased.

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20 Expressing Probability Lesson 6.2.2 Round Up Counting outcomes is fairly straightforward in simple situations like these. But for more complicated sets of events and outcomes, you’ll need to organize your information.

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