Download P.o.D. 1.) In how many ways can a 12 question true

P.o.D.
1.) In how many ways can a 12
question true-false quiz be
answered?
2.) How many 4-letter, 3-number
license plates be created?
3.) In how many ways can the
letters in PRECALCULUS be
arranged?
4.) MISSISSIPPI
1.) 4,096
2.) 26(26)(26)(26)(10)(10)(10)=
456,976,000
3.)
11!
2!2!2!
= 4,989,600
4.)
11!
4!4!2!
= 34,650
9.7 – Probability
Learning Target: I can find the
probabilities of events
Probability – if an event can
succeed in s ways and fail in f
ways, then its probability is
𝑠
(
)
𝑃 𝑠 =
.
𝑠+𝑓
EX: A bag contains 3 red balls
and 5 green balls. If one ball is
selected at random what is the
probability that it is red?
3
3
=
3+5 8
What is the probability that it is
green?
5
8
Notice that probabilities must
3
5
8
8
8
8
always add to 1: + = = 1
EX: One bag of gummy bears
contains 15 red, 10 yellow, and 6
green. Find the probability of
each selection.
a.)
b.)
c.)
d.)
One red
Not picking a yellow
Picking one green
Not picking a red
a.)
15
31
b.)
c.)
d.)
15+6
31
6
31
10+6
31
=
=
21
31
16
31
Odds – the ratio of successful
outcomes to failures
EX: What are the odds of
drawing a 5 from a standard
deck of 52 cards?
How many five’s are in the deck?
4
How many cards are not five’s?
48
So the odds are 4:48 or 1:12. This
can also be expressed as “1 to 12”
or
1
.
12
EX: On an ACT Review Guide,
there are 5 math questions and 4
science questions. Arthur picks
a question at random to answer.
What are the odds of not picking
a science question?
5:4
EX: What are the odds that a
person chosen at random got a
passing grade on an algebra test
if the scores were 3 A’s, 4 B’s, 10
C’s, 2 D’s, and 2 F’s?
19:2
Probability of Two Independent
Events:
If two events A and B are
independent, then the
probability of both events
occurring is found by
P(A and B)=P(A) P(B)
EX: A bag contains 5 red marbles
and 4 white marbles. A marble is
to be selected and replaced in
the bag. A 2nd selection is then
made. What is the probability of
selecting 2 red marbles?
5 5
25
( )=
9 9
81
EX: A box contains 5 triangles, 6
circles, and 4 squares. If a figure
is removed, replaced, and a
second is picked, what is the
probability that a triangle and
then a circle will be picked?
5 6
1 2
2
( )= ( )=
= 0.1333
15 15
3 5
15
= 13.3%
EX: A jar contains 7 lemon
gumdrops, 3 cherry gumdrops,
and 8 rainbow gumdrops. What
is the probability of selecting 2
lemon in succession providing
each gumdrop is replaced?
7 7
49
( )=
= 0.151 = 15.1%
18 18
324
Probability of Two Dependent
Events:
If 2 events A and B are
dependent, then the probability
of both events occurring is
found by
P(A and B)=P(A)P(B following A)
EX: There are 7 dimes and 9
pennies in a purse. Suppose two
coins are to be selected at
random, without replacing the
first one. Find the probability of
picking a penny and then a dime.
9 7
63
21
( )=
=
= 0.2625
16 15
240 80
= 26.25%
EX: What is the probability of
drawing 2 cards showing odd
numbers from a set of cards that
show the first 20 counting
numbers if the 1st card is not
replaced?
10 9
90
9
( )=
=
= 0.2368
20 19
380 38
= 23.68%
EX: There are 3 quarters, 4
dimes, and 7 nickels in a change
purse. Suppose 3 coins are to be
selected without replacement.
What is the probability of
selecting a quarter, then a dime,
and then a nickel?
3 4
7
84
1
( )( ) =
=
= 0.03846
14 13 12
2184 26
= 3.85%
Mutually Exclusive – events that
cannot occur at the same time
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Inclusive Events – two events
that can occur at the same time
𝑃(𝐴 𝑜𝑟 𝐵)
= 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
EX: What is the probability of
drawing a jack or a king from a
standard deck of cards?
Exclusive
4
4
8
2
𝑃(𝑗𝑎𝑐𝑘 𝑜𝑟 𝑘𝑖𝑛𝑔) =
+
=
=
52 52 52 13
EX: A card is selected from a
standard deck of cards. What is
the probability that it is a red
card or an ace?
Inclusive
𝑃(𝑟𝑒𝑑 𝑜𝑟 𝑎𝑐𝑒)
= 𝑃(𝑟𝑒𝑑) + 𝑃(𝑎𝑐𝑒) − 𝑃(𝑟𝑒𝑑 𝑎𝑛𝑑 𝑎𝑐𝑒)
26 4
2
28
7
=
+
−
=
=
52 52 52 52 13
EX: 3 cards are selected from a
standard deck. What is the
probability of selecting a king, queen,
or a red card?
Inclusive
𝑃(𝑘𝑖𝑛𝑔, 𝑞𝑢𝑒𝑒𝑛, 𝑜𝑟 𝑟𝑒𝑑 )
4
4 26 2
2
30 15
=
+
+
−
−
=
=
52 52 52 52 52 52 26
EX: An Easter basket contains 45 dyed
eggs: 15 yellow, 12 green, and 18 red.
What is the probability of selecting a
green or red egg?
Exclusive
12 18 30 2
𝑃(𝑔𝑟𝑒𝑒𝑛 𝑜𝑟 𝑟𝑒𝑑) =
+
=
=
45 45 45 3
EX: The letters from the word LOVE
and LIVE are placed on cards and put
in a box. What is the probability of
selecting an L or O from the box?
Inclusive
2 1 3
𝑃(𝐿 𝑜𝑟 𝑂) = + =
8 8 8
EX: The letters of the alphabet are
placed in a bag. What is the
probability of selecting a vowel or the
letters QUIZ?
5
4
2
7
𝑃(𝑣𝑜𝑤𝑒𝑙 𝑜𝑟 𝑄𝑈𝐼𝑍) =
+
−
=
26 26 26 26
EX: Two coins are tossed. What is the
probability that one will land heads
up and the other will land heads
down?
Possibilities: HH, HT, TH, TT
2 1
=
4 2
EX: A card is drawn from a standard
deck of playing cards. What is the
probability that the card is a heart?
1
4
EX: A 6-sided die is tossed twice. What
is the probability that the total of the
2 tosses is 5?
Ways to add to 5:
1,4
2,3
3,2
4,1
There are 6(6)=36 possible outcomes.
4
1
=
36 9
EX: A card is selected from a standard
deck. What is the probability that the
card is either a club or an ace?
13 4
1
16
4
𝑃(𝑐𝑙𝑢𝑏 𝑜𝑟 𝑎𝑐𝑒) =
+
−
=
=
52 52 52 52 13
EX: Two integers from 1 to 30 are
chosen by a random number
generator. What is the probability
that both numbers are less than 12?
11 11
121
( )=
30 30
900
EX: A sales representative makes a
sale at approximately one-third of all
calls. If, on a given day, the
representative contacts five potential
clients, what is the probability that a
sale will be made with each of the five
contacts?
1 5
1
( ) =
3
243
EX: A sales rep makes a sale at
approximately one-third of all calls.
If, on a given day, the representative
contacts 5 potential clients, what is
the probability that a sale will be
made with AT LEAST one contact?
*We want to find the complement. In
other words, what is the likelihood
that no sale occurs?
2 5
32
𝑃(𝑛𝑜 𝑠𝑎𝑙𝑒) = ( ) =
3
243
32
211
𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑠𝑎𝑙𝑒) = 1 −
=
243 243
Upon completion of this lesson, you
should be able to:
1. Determine basic probabilities.
2. Calculate odds.
3. Differentiate between
independent and dependent
events.
a. And find their probabilities.
4. Differentiate between inclusive
and exclusive events.
a.
And find their probabilities.
For more information, visit
https://www.youtube.com/watch?v=qGiKrM
CYCiI
HW
Pg.709
3-48 3rds
Quiz 9.5-9.7 tomorrow