Transcript Probability of an Event Not Happening Lesson 6.2.4 Lesson 6.2.4 Probability of an Event Not Happening California Standard: What it means for you: Statistics, Data Analysis and Probability 3.3 Represent.

Probability of an Event
Not Happening
Lesson 6.2.4
1
Lesson
6.2.4
Probability of an Event Not Happening
California Standard:
What it means for you:
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.
You’ll learn how to find the
probability that a particular
event does not happen.
Key words:
•
•
•
•
probability
event
outcome
favorable
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Lesson
6.2.4
Probability of an Event Not Happening
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.
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Lesson
6.2.4
Probability of an Event Not Happening
Find Probabilities by Counting Outcomes
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
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Lesson
6.2.4
Example
Probability of an Event Not Happening
1
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?
Solution
There are 3 possible outcomes:
1
red, yellow, and blue. So P(Yellow) =
3
If he doesn’t spin yellow, Ruben must spin either red or blue.
So there are 2 favorable outcomes for not spinning yellow.
2
So P(Not yellow) = P(Red or blue) =
5
3
Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
Guided Practice
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 1
=
6 3
3. P(Yellow)
1
6
5. P(Red)
2 1
=
6 3
7. P(Orange)
0
2. P(Not green)
4 2
=
6 3
4. P(Not yellow)
5
6
6. P(Not red)
4 2
=
6 3
8. P(Not orange) 1
6
Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
You Can Find P(not A) by Counting Outcomes
You can use the probability that event A will happen to find
the probability that A won’t happen.
Example 2 shows you how.
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Lesson
Probability of an Event Not Happening
6.2.4
Example
1
3
7
5
2
Mario is playing a game using this spinner.
He spins it twice, and adds the numbers he
spins to get his score.
1st spin
If A is the event “Mario scores 8,”
then find: (i) P(A), (ii) P(not A).
2nd spin
This table gives the possible outcomes.
+
1
3
5
7
1
2
4
6
8
3
4
6
8 10
5
6
8 10 12
7
8 10 15 14
What do you notice about P(A) + P(not A)?
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Solution follows…
Lesson
6.2.4
Example
Probability of an Event Not Happening
2
1st spin
2nd spin
If A is the event “Mario scores 8,”
then find: (i) P(A), (ii) P(not A).
Solution
+
1
3
5
7
1
2
4
6
8
3
4
6
8 10
5
6
8 10 12
7
8 10 15 14
(i) The table shows there are 16 possible outcomes.
There are 4 possible outcomes where Mario does score 8.
4 1
So P(Mario scores 8) = = (or 25%).
16 4
9
Solution continues…
Lesson
6.2.4
Example
Probability of an Event Not Happening
2
1st spin
2nd spin
If A is the event “Mario scores 8,”
then find: (i) P(A), (ii) P(not A).
Solution (continued)
+
1
3
5
7
1
2
4
6
8
3
4
6
8 10
5
6
8 10 12
7
8 10 15 14
(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.
12 3
So P(Mario does not score 8) = = (or 75%).
16 4
10
Solution continues…
Lesson
6.2.4
Example
Probability of an Event Not Happening
2
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)?
Solution (continued)
1
3
P(A) = (or 25%) and P(not A) = (or 75%)
4
4
1 3
So, P(A) + P(not A) = + = 1 (or 100%).
4 4
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Lesson
6.2.4
Probability of an Event Not Happening
You Can Also Find P(not A) If You Know P(A)
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.
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Lesson
6.2.4
Example
Probability of an Event Not Happening
3
The weather channel says there is 30% chance of rain today.
What is the probability that it will not rain today?
Solution
P(rain) + P(not rain) = 100%
So 30% + P(not rain) = 100%
This means P(not rain) = 100% – 30% = 70%
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Solution follows…
Lesson
Probability of an Event Not Happening
6.2.4
Guided Practice
Exercises 9–16 give P(A). Find P(not A) in each case.
3
9. P(A) =
10. P(A) = 0.58
4
1 – 0.58 = 0.42
1–
3 1
=
4 4
11. P(A) = 11%
100% – 11% = 89%
13. P(A) = 0.9
1 – 0.9 = 0.1
12. P(A) = 43%
100% – 43% = 57%
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14. P(A) =
12
1–
11 1
=
12 12
15. P(A) = 1
16. P(A) = 0
1–1=0
1–0=1
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Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
Guided Practice
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)
1st spin
18. P(Mario doesn’t score 14)
1 – 0.0625 = 0.9375
19. P(Mario scores 6)
3 ÷ 16 = 0.1875
2nd spin
1 ÷ 16 = 0.0625
+
1
3
5
7
1
2
4
6
8
3
4
6
8 10
5
6
8 10 12
7
8 10 15 14
20. P(Mario doesn’t score 6)
1 – 0.1875 = 0.8125
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Solution follows…
Lesson
Probability of an Event Not Happening
6.2.4
Guided Practice
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)
1st spin
22. P(Mario doesn’t score less than 10)
1 – 0.625 = 0.375
23. P(Mario scores 7)
0 ÷ 16 = 0
2nd spin
10 ÷ 16 = 0.625
+
1
3
5
7
1
2
4
6
8
3
4
6
8 10
5
6
8 10 12
7
8 10 15 14
24. P(Mario doesn’t score 7)
1–0=1
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Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
Guided Practice
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.
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Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
Independent Practice
1. The probability of an event occurring is 0.6.
Explain why the probability of the event not occurring
cannot be 0.2.
The probabilities of an event occurring and not occurring must
add up to 1. 0.6 + 0.2 = 0.8, and not 1.
2. P(A occurring) and P(A not occurring) are the same.
Find P(A).
P(A) + P(not A) = 1, and P(A) = P(not A)  P(A) =
1
2
3. P(A occurring) is twice P(A not occurring). Find P(A).
P(A) + P(not A) = 1, and P(A) = 2 × P(not A)  P(A) =
2
3
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Solution follows…
Lesson
Probability of an Event Not Happening
6.2.4
Independent Practice
Cynthia picks one card from a standard pack of 52D cards.
= diamonds,
Card 1
spades,
She makes a note of the suit, then replaces the cardS =and
picks
C = clubs,
another one. The tree diagram shows the possible outcomes.
H = hearts,
Card 2
D = diamonds,
S = spades,
C = clubs,
Outcomes DD DC DH DS CD CC CH CS HD HC HH HS SD SC SH SS
H = hearts,
Card 2
Card 1
Calculate P(A) and P(not A), where event A is Cynthia picking:
4. At least one heart
5. Two diamonds
7 DD DC DH DS9 CD CC CH CS HD HC HH HS SD
1 SC SH SS
Outcomes
P(A) =
16
, P(not A) =
16
P(A) =
16
, P(not A) =
15
16
6. At least P(A)
one red
OneAheart
and one
club
Calculate
andcard
P(not A), where 7.
event
is Cynthia
picking:
1
7
3
1
4. At
least
5. Two
P(A)diamonds
= , P(not A) =
P(A)
= , one
P(notheart
A) =
8
8
4
4
6.
At
least
one
red
card
7.
One
heart
and
one
club
8. Two cards the same suit
1
3 suit
19
8. Two
cards
the
same
P(A) = , P(not A) =
4
4
Solution follows…
Lesson
6.2.4
Probability of an Event Not Happening
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.
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