1 Lesson 6.2.2 Expressing Probability. 2 Lesson 6.2.2 Expressing Probability California Standard:...

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Lesson 6.2.2Lesson 6.2.2

Expressing ProbabilityExpressing Probability

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Lesson

6.2.2Expressing ProbabilityExpressing 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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Expressing ProbabilityExpressing ProbabilityLesson

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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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) = .”12

12

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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 outcomesNumber of possible outcomes

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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) = 16

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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

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250 out of the 100 marbles are red, so P(red) = =

50

1003

1030 out of the 100 marbles are blue, so P(blue) = =

30

1001

520 out of the 100 marbles are green, so P(green) = =

20

100

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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

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225 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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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

521

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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

6P(odd) = =


1

2

3

6P(even) = =

There are 4 favorable outcomes (3, 4, 5, 6) out of 6 possible outcomes.

2

3

4

6P(>2) = =


1

2

3

6P(prime) = =

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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

52P(JD) =

There are 4 favorable outcomes (7D, 7C, 7H, 7S) out of 52 possible outcomes.

4

52P(7) = =

1

13

There are 13 favorable outcomes (AC – KC) out of 52 possible outcomes.

13

52P(C) = =

1

4

There are 26 favorable outcomes out of 52 possible outcomes.

26

52P(black) = =

1

2

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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 outcomesNumber of possible outcomes

To do this, you need to rearrange this formula:

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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 outcomesNumber of possible outcomes

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= Number of favorable outcomes

2014

Number of favorable outcomes = 20 × = 5 14

Multiply both sides by 20There are 20

marbles in total

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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?

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Number of marbles = 11 × 2 = 2211

Number of marbles=

1

2

Number of red marbles = 100 ÷ 10 = 10Number of red marbles

10010% = =

1

10

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Guided Practice

Solution follows…

Lesson

6.2.2


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?

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Number of marbles25% = =

1

4


Number of marbles=

1

3

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Guided Practice

Solution follows…

Lesson

6.2.2


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%?

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Number of marbles20% = =

1

5

Number of silver marbles = (3 × 80) ÷ 4 = 60Number of silver marbles

80=

3

4

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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

80

1

20=

15

80=

3

16

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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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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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6.2.2

Round UpRound 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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