# Probability by jennyyingdi

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1. Is counting the number of ways to form a 3 person
committee out of 10 people a combination or a permutation?

2. How many different licence plates are possible if there
are 2 letters, followed by 4 numbers, then 2 letters?
3. What is the probability of dealing a king out of a well-shuffled
deck of cards?

4. If P(A) = 0.5, P(B) = 0.4, and P(A or B) = 0.8. Are the two
events A and B, disjoint or overlapping? What is P(A and B)?

5. How many ways can you arrange 10 candidates into the
positions of President, Vice President, and Secretary ?
10-5
Conditional and Unconditional
Probability.

(Sequential events)
The Probability of Sequential Events
The probability of event A happening followed by event B.

Event A: rolling a 5 on a single die.
Event B: rolling a 6 on a single die.

Event A: drawing a red marble out of a bag of marbles
(containing more than one red marble and some other colors)
Event B: drawing another red marble out of the same bag of
marbles (without replacement)

How are these two situations different?
Vocabulary
Conditional Probability: the probability of an event depends
upon the outcome of a previous event occurring (drawing a
red ball on the second draw out of a bag of colored
marbles).

Unconditional Probability : the probability of an event does
not depend upon the outcome of a previous event occurring
(the probability of drawing a king on the second card dealt
when the previously dealt card was shuffled back
into the deck before drawing the second card (with
replacement).
Conditional vs. Unconditional Probability.
150 raffle tickets were sold for a mall gift certificate. 200
tickets were sold for a movie pass.
What is the probability that you win both?

Does the probability of winning the mall gift certificate depend
upon whether or not the movie pass was won or not?

Winning the gift certificate or winning the movie pass are
independent events.
Dependent vs. Independent Events
A BMX bicycle race as 8 starting positions called “slots”. What
is the probability of drawing slot #8 for 3 races is a row?

Does the probability of drawing slot #8 on the second race
depend upon the slot that was drawn in the previous race?

Drawing a specific slot for subsequent races are
independent events.
Dependent vs. Independent Events
Which is it, dependent or independent?
For fund raiser, 100 tickets are sold for dinner for two at Chile’s
Restaurant, and 200 tickets are sold for a booklet of movie
passes. You buy 3 of the dinner and 5 of the movie raffle tickets.
What is the probability of winning both?

What is the probability of rolling a pair of dice and having their
sum be 7 ?
Your turn: which is it, dependent or independent?

1. For fund raiser, 100 tickets are sold for dinner for two at at
Chile’s Restaurant, and 200 tickets are sold for a booklet of movie
passes. You buy 3 of the dinner and 5 of the movie raffle tickets.
What is the probability of winning the movie booklet?
2. A red marble is drawn from a bag of marbles. What is the
probability of drawing another red marble (without
replacement) ?
3. A card is dealt face up from a deck of cards. What is the
probability that the second card dealt will be a king (without
replacing the first card)?
4. What is the probability of rolling a 5 then rolling another 5
using the same die?
Probability of Independent Events
P(A and B) where events “A” and “B” are independent:

P(A and B) = P(A) * P(B)

What is the probability of rolling a 5 then rolling another
5 using the same die?

1 1           1
P (“5” and “5”) = ?    *               0.027
6 6           36
Independent Probability.
For fund raiser, 100 tickets are sold for dinner for two at at Chile’s
Restaurant, and 200 tickets are sold for a booklet of movie
passes. You buy 3 of the dinner and 5 of the movie raffle tickets.
What is the probability of winning both?

3   5     15
P (“dinner” and “movie”) = ?     *    
100 200   20000

3
       0.00075
4000
Independent Probability.

5. What is the probability of rolling “snake eye’s” (two 1’s)
followed by another “snake eye’s” on the second roll of a
pair of “fair” dice ?

6. Events “A” and “B” are independent events.
P(A) = 0.9
P(B) = 0.8
P(A and B) = ?
Geometric Probability:
Assumming that at an arrow randomly hits anywhere in the four
square area, what is the probability of hitting in the #1 square
with the first dart and the #4 square with the second dart?

What would be the
1           2            probability of hitting in order
squares #1 through #4 with
4 darts thrown one after the
other?

3             4
Probability
7. What is the probability of hitting the outer ring on with the
first dart and the “bull’s eye” with the second dart?

8. What is the probability hitting the “bull’s eye” on the first try
then not hitting the “bull’s eye” on the second try?

3
2


Dependent Probability.
A card is dealt face up from a deck of cards. What is the
probability that the second card dealt will be a king?
The probality of getting a king on the second card depends
upon whether there was a king drawn on the first card.

Case 1: king was drawn               Case 2: king not drawn
first (without                       first (no replacement
replacement)                         of the card)
3                               4
P(2 king) 
nd
P(2 nd king) 
51                               51
The probability of the second event depends upon whether
there was a king drawn on the first card.
Probability of Dependent Events
P(A and B) where events “A” and “B” are dependent:

P(A and B) = P(A) * P(‘B’ given that event ‘A’ occurred)
P(A and B)  P( A) * P B A
Event B: rolling a 6 on a single die.
A card is dealt face up from a deck of cards. What is the
probability that the first card is not a king and the second
card dealt is a king?
48                                                        4
P(1 not a king) 
st                       P(2nd is a king given that 1st as not a king) 
52                                                        51

P(A and B)  P( A) * P B A          48 4
 *
52 51
192
          0.072
2652
Probability of Dependent Events
P(A and B) = P(A) * P(‘B’ given that event ‘A’ occurred)

A bag contains 3 red, 3 blue, and 3 green marbles. You and two friends
pick a marble from the bag (without replacing). What is the
probability that you will each have a different color of marble?
# of ways to choose a marble                      9
First draw:    P(event )                                                         
total # of ways that 9 marbles can be chosen               9

Second P(event)  # of ways to choose one marble out of 2 colors                
6
draw:              total # of ways that 8 marbles can be chosen                     8

# of ways to choose one marble out of 1 color       3
Third      P(event)                                                    
total # of ways that 7 marbles can be chosen        7
draw:
9 6 3               18
P(event)  * *                   0.321
9 8 7               56
Independent Probability.
A bag of marbles contains 3 red, 4 blue, and 6 green marbles.
3
What is the probability of drawing a red marble   1st?   
13
10
What is the probability of not drawing a red marble   1st?   
13
What is the probability of drawing a red marble 1st, then not
drawing a red marble 2nd (with replacement)?
Can I use this equation?     P(A and B)  P( A) * P B A

3 10          30
 *                0.18
13 13         169
9. What is the probability of drawing an ace, then (without
replacing the ace) drawing a king out of well-shuffled deck
of cards?
10. Events “A” and “B” are dependent events.
P(A) = 0.5
P(B given that A occured) = 0.4
P(A and B/A) = ?
11. The probability of dealing a heart on the first card is:
P(heart) = 0.25

What is the probability of drawing a heart 1st followed by not
drawing a heart on the 2nd card (no replacement)?
Total possible outcomes of rolling 2
dice: P( rolling a sum of 5 using 2 dice) = ?
die roll     1       2       3       4        5        6

1       1+1=2   2+1=3   3+1=4   4+1=5    5+1=6    6+1=7

2       1+2=3   2+2=4   3+2=5   4+2=6    5+2=7    6+2=8

3       1+3=4   2+3=5   3+3=6   4+3=7    5+3=8    6+3=9

4       1+4=5   2+4=6   3+4=7   4+4=8    5+4=9    6+4=10

5       1+5=6   2+5=7   3+5=8   4+5=9    5+5=10   6+5=11

6       1+6=7   2+6=8   3+6=9   4+6=10   5+6=11   6+6=12

4/36 = 1/9 = 0.1111

12. What is the probability of having the sum of two fair dice
being a 7?

13. What is the probability of being dealt a queen then an ace
if the queen is put back into the deck and shuffled before
you are dealt the second card (with replacement)?

14. What is the probability of being dealt a queen then an ace
if the queen is not put back into the deck before you are
dealt the second card (without replacement)?
Probability
In the movie “Tron” a game is played where two contestants
are on circular stands. One player throws a ball toward the
player’s circle. If that player doesn’t catch the ball, it will hit
one of the rings he is standing on and the ring will disappear.
He needs to either catch the ball or get off the ring before it
disappears. If a player is standing on a ring when it disappears
he will fall to his death.
125
15. What is the probability of a                   75
player still being alive if he
doesn’t catch the ball, doesn’t
25
move and he is standing on
the blue ring?
Dependent/independent vs.
Disjoint/overlapping events.
Dependent/Independent events deal with sequential events.
Drawing a king then a 5.
P(A and B)  P( A) * P B A
Drawing a red then a blue marble

Disjoint/overlapping events deal with an event that can be
categorized in more than one way.

Being a girl with brown hair in a group of boys and
girls of different hair colors.
P(A or B)= P(A) + P(B) – P(overlap)
Summary
Disjoint/overlapping events deal with an event that can be
categorized in more than one way.
P(boy or girl)  ?      P(A or B)= P(A) + P(B) – P(overlap)
(Blonde           (Girls)
2 3 0        5
(boys)Hair)
Amber        Maria              
Angelica    5 5 5        5
Bill

2 2 1         3
P(boy or blond hair)  ?           
Frank                                5 5 5         5

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