Transcript 5.2B Multiplication Rules

5.2B Multiplication Rules
Independent Events
Dependent Events
General Multiplication Rule
Independent and Dependent
Events
Independent Events:
Two events are independent if knowing
that one will occur (or has occurred) does
not change the probability that the other
occurs.
Independent and Dependent
Events
Dependent Events:
Two events are dependent if knowing that
one will occur (or has occurred) changes
the probability that the other occurs.
Example #1
The following are examples of independent
events:
a. Rolling a die AND getting a 6, and
then rolling a second die and getting a
3.
b. Drawing a card from a deck AND
getting a queen, replacing it, then
drawing a second card and getting a
king.
Example #1
The following are examples of independent
events:
c. Being on time to school AND your
teacher being on time to school.
d. Choosing a marble from a jar AND
tossing a coin that lands on heads.
Example #2
The following are examples of dependent
events:
a. The speed you drive to school AND the
weather.
b. Choosing a marble from a jar, not
replacing it, AND drawing another
marble from that same jar..
Example #2
The following are examples of dependent
events:
c. Eating a full breakfast AND being on
time to school.
d. Parking in a no-parking zone AND
getting a parking ticket.
Example #3
Determine whether the events are
independent or dependent.
a. Tossing a coin and drawing a marble out
of a bag.
INDEPENDENT
b. Eating sweets and having diabetes.
DEPENDENT
Example #3
Determine if the events are independent or
dependent.
c. Being on the Indianapolis Colts football
team and being a winner
DEPENDENT
d. Drawing a king from a standard deck,
replacing it and drawing another king.
INDEPENDENT
Multiplication Rule For
Independent Events
If events A and B are independent,
P A and B  P A PB
Example #4
A dresser drawer contains one pair of socks
of each of the following colors: blue, brown, red,
white and black. Each pair is folded together in
matching pairs. You reach into the sock drawer
and choose a pair of socks without looking. The
first pair you pull out is red -the wrong color. You
replace this pair and choose another pair. What is
the probability that you will choose the red pair
of socks twice?
Example #4
Indepdendent?
Yes
P( R and R)  P ( R )  P ( R ) 
 1  1 
    .04
 5  5 
Example #5
A coin is tossed and a single 6-sided die is
rolled. Find the probability of landing on
the head side of the coin and rolling a 3 on
the die.
Independent?
Yes
Example #5
PH and 3  PH  P3 
 1  1  1
 .083
   
 2  6  12
Example #6
A card is chosen at random from a deck of
52 cards. It is then replaced and a second
card is chosen. What is the probability of
choosing a face card and an eight?
Independent?
Yes
Example #6
PFace and 8  PFace P8
 12  4 
    .018
 52  52 
Example #7
A South Carolina survey of registered voters
found that 65% were opposed to the new
Health Care Plan. Suppose you randomly
choose 5 South Carolinians. What is the
probability all 5 of them oppose the health
care plan?
Independent?
Yes
Example #7
P3 oppose  PO and O and O 
.65.65.65  .65
3
 .275
General Multiplication Rule
Given events A and B, the probability of
both A and B occurring is:
P(A and B) = P(A)P(B|A),
Where P(B|A) is the probability that B
occurs given A has occurred.
Example #8
A card is chosen at random from a standard
deck of 52 playing cards. Without replacing
it, a second card is chosen. What is the
probability that the first card chosen is a
queen and the second card chosen is a
jack?
Independent?
No
Example #8
PQ and J   PQ PJ | Q 
 4  4 
    .00603
 52  51 
Example #9
Mr. Parietti needs two students to help him
with a science demonstration for his class of
18 girls and 12 boys. He randomly chooses one
student who comes to the front of the room.
He then chooses a second student from those
still seated. What is the probability that both
students chosen are girls?
Independent?
No
Example #9
P2G  PG PG | G 
 12  11 
    .152
 30  29 
Example #10
In a shipment of 20 computers, 3 are
defective. Three computers are randomly
selected and tested. What is the
probability that all three are defective if
the first and second ones are not replaced
after being tested?
Independent?
No
Example #10
P3 Defective  PD PD | 1D PD | 2D 
 3  2  1 
     .00088
 20  19  18 
Example #11
On a math test, 5 out of 20 students got an
A. If three students are chosen at random
without replacement, what is the
probability that all three got an A on the
test?
Independent?
No
Example #11
P3 A' s   P A P A | 1A P A | 2 A' s  
 5  4  3 
     .0088
 20  19  18 
Example #12
A jar contains 6 red balls, 3 green balls, 5
white balls and 7 yellow balls. Two balls
are chosen from the jar, with replacement.
What is the probability that both balls
chosen are green?
Independent?
Yes
Example #12
P2G  PG PG 
 3  3 
    .0204
 21  21 
Example #13
A nationwide survey showed that 73% of all
children in the United States dislike eating
vegetables. If 5 children are chosen at
random, what is the probability that all 5
dislike eating vegetables?
Independent?
Yes
Example #13
P5 Dislike Veggies   PDislike Veggies  
5
.73  .207
5
Example #14
A school survey found that 7 out of 30
students walk to school. If four students
are selected at random without
replacement, what is the probability that
the first two chosen walk to school and the
next two do not walk to school?
Independent?
No
Example #14
P2 Walk , 2 Don ' t Walk  
PW   PW | W   PDW | 2W   P( DW | 2Wand1DW ) 
 7  6  13  12 
      .056
 20  19  18  17 