easy paper
snoopy beats
h__3._ch._3.pdf
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Probability Ch. # 3
3.1 History of Probability Objectives
Understand the types of problems that motivated the invention of probability theory.
Become familiar with dice, cards, and the game of roulette.
The American roulette wheel has thirty-
eight numbered compartments around its
circumference.
Thirty-six of these compartments are
numbered from 1 to 36 and are colored
red or black.
The remaining two are numbered 0 and 00
and are colored green. Players place their
bets by putting their chips on an
appropriate spot on the roulette table
The dealer spins the wheel and then drops
a ball onto the spinning wheel. The ball
eventually comes to rest in one of the compartments, and that compartment’s number is the
winning number.
For example, if a player wanted to bet $10 that the ball lands in compartment number 7, she
would place $10 worth of chips on the number 7 on the roulette table.
This is a single-number bet, so house odds are 35 to 1
This means that if the player wins, she wins $10 35 = $350, and if she loses, she loses her $10.
Similarly, if a player wanted to bet $5 that the ball lands either on 13 or 14, he would place $5
worth of chips on the line separating numbers 13 and 14 on the roulette table. This is a two-
numbers bet, so house odds are 17 to 1.
This means that if the player wins, he wins $5 17 = $85, and if he loses, he loses his $5.
A modern deck of cards contains fifty-two cards (thirteen in each of four suits). The four suits
are hearts, diamonds, clubs, and spades ( ). Hearts and diamonds are red, and clubs
and spades are black.
Each suit consists of cards labeled 2 through 10, followed by jack, queen, king, and ace. Face
cards are the jack, queen, and king; and picture cards are the jack, queen, king, and ace.
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3.2 Basic Terms Of Probability Objectives
Learn the basic terminology of probability theory.
Be able to calculate simple probabilities.
Understand how probabilities are used in genetics.
Experiment:
A process by which an observation, or outcome, is obtained.
Sample Space:
The sample space of an experiment is the set S of all outcomes.
A Certain Event:
Is an event that is certain to happen, it is an event that’s equal to the sample space S.
An impossible event:
Is an event that cannot happen, it is an event that is equal to the empty set
Equally Likely Outcomes:
If each outcome of an experiment has the same chance of occurring as any other
outcomes, then they are said to be equally likely outcomes.
A coin is considered fair if the outcomes of head and tail are equally likely.
Mutually Exclusive
Two events E and F are said to be mutually exclusive if: E F
Ex:
A die is rolled. Let E be the event “an even number comes up,” F the event “a number
greater than 3 comes up,” and G the event “an odd number comes up.”
a. Are E and F mutually exclusive? b. Are E and G mutually exclusive?
Solution
a. 2, 4, 6 , 4, 5, 6E F and 4, 6E F Therefore, E and F are
not mutually exclusive; the number that comes up could be both
even and greater than 3. In particular, it could be 4 or 6.
b. 2, 4, 6 , 1, 3, 5E G And E G . Therefore, E and G are
mutually exclusive; the number that comes up could not be both
even and odd.
Probability
If an experiment can result in any of (n ≥ 1) mutually exclusive and equally likely
outcomes, and if s of these are consider favorable to event E, then:
Odds in favor of an event E:
The odds of an event E with equally likely outcomes, denoted by:
/ n E
O E n E
( )
n Esuccess P E
total n S
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Relative Frequency:
If an experiment is repeated n-times and an event occurred m-times, and then number of times the outcomes occursm
n total number of repetitions are called the relative frequency of the event.
Law of Large Numbers:
If an experiment is repeated a large number of times, the relative frequency of an
outcome will tend to be close to the probability of that outcome.
A gene is a biological determiner of some inherited quality in a living being.
Genes come in pairs, with one gene coming from one parent and the other gene from the
other parent.
A dominant gene is one that determines a particular inherited quality regardless of what gene it
is paired with.
A recessive gene is one that determines a particular inherited quality only if it is paired with
another gene that’s exactly the same.
Ex:
1. In Mendel’s experiment, we use R to stand for the dominant red gene and w (lowercase) to stand for recessive white gene.
When Mendel’s cross-fertilized pure-red plants with pure-white plants, the offspring all
had red flowers.
R R First parent’s genes
w (R,w) (R,w) Possible offspring
w (R,w) (R,w) Possible offspring
A Punnett square for the first generation.
When these first-generation offspring were cross-fertilized, Mendel’s found that
approximately ¾ of the resulting second generation offspring had red flowers and ¼ had
white flowers.
R w First parent’s genes
w (R,w) (w,R) Possible offspring
w (R,w) (w,w) Possible offspring
A Punnett square for the second generation.
The sample space for the second generation: , , , , , , ,S R R R w w R w w ,
, , , , ,RE R R R w w R , because R dominates w.
,wE w w ,
3
4
R
R
n E P E
n S
1
4
w n E
P Ew n S
4
Ex:
2. A coin is flipped. Find the following. a. The sample space b. The probability of event E1, “getting heads” c. The odds of event E1, “getting heads” d. The probability of event E2, “getting heads or tails” e. The odds of event E2, “getting heads or tails”
Solution
a. Finding the sample space S: The experiment is flipping a coin. The only possible outcomes are heads and tails.
The sample space S is the set of all possible outcomes, so S = {h, t}.
b. Finding the probability of heads: E1 = {h} (“getting heads”)
1
1
1
2
n E P E
n S
This means that one out of every two possible outcomes is a success.
c. Finding the odds of heads: E1 = {t}
o(E1) = n(E1) : n(E1 ) = 1:1
This means that for every one possible success, there is one possible failure.
d. Finding the probability of heads or tails:E2 = {h, t}
2
2
2 1
2 1
n E P E
n S
This means that every outcome is a success. Notice that; E2 is a certain event.
e. Finding the odds of heads or tails: E2 = Ø
o(E2) = n(E2) : n(E2 ) = 2:0 = 1:0
This means that there are no possible failures.
3. Use probability rules to find the probability that a child will have neither sickle-cell anemia nor sickle-cell trait if the following occurs. (Enter your answers as fractions.)
a. each parent has sickle-cell trait b. one parent has sickle-cell trait and the other parent has no sickle-cell gene c. one parent has sickle-cell anemia and the other parent has no sickle-cell gene
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3.3 Basic Rules of Probability Objectives
Understand what type of number a probability can be
Learn about the relationships between probabilities, union, and intersections.
Probability Rules
1. ( ) 0( ) 0 ( ) ( )
n P
n S n S
(impossible event)
2. ( )( ) 1 ( )
n S P S
n S (certain event)
3. 0 ( ) ( )
0 ( ) ( )
( ) ( ) ( )
0 ( ) 1
n E n S
n E n S
n S n S n S
P E
4. The Union/ Intersection Rule: ( ) ( ) ( )P E F P E P F P E F
5. Mutually exclusive Rule: ( ) ( )P E F P E P F
6. The Complement Rule: /
( ) ( ) 1P E P E
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3.4 Basic Rules of Probability
Objectives
Apply the concepts of permutations and combinations to probability calculations
Use probabilities to analyze games of chance such as the lottery.
Which Combinatory Method?
1. If the selection is done with replacement, use the fundamental Principle of Counting. 2. If the selection is done without replacement and there is only one category, use:
a. permutation if the order of the selection does matter:
n r
! P
!
n
n r
b. combination if the order of the selection does not matter: !
( )! ! n r
n C
n r r
3. If there is more than one category, use the Fundamental Principle of Counting.
Ex:
1. Consider a box containing marbles: 25 black, 30 blue, 15 red and 10 white. Find:
( ), ( ), ( )P B P W and P R
2. Find the probability for this experiment by rolling a pair of dice. P (nine), P (odd), P (even).
3. What is the sample space for the experiment of tossing a coin and rolling a die? S = { H1, H2, …, H6, T1, T2, …, T6}
4. Consider two spinners, you spin, the one with higher number wins. Which spinner should you choose? Explain why?
D
E 4 5 6
1 (1,4) (1,5) (1,6)
7 (7,4) (7,5) (7,6)
8 (8,4) (8,5) (8,6)
( ) ( )P D P E 5. Suppose you toss a coin and roll a die. Find: ( 1), ( 3)P H and P T or
6. Suppose pair of dice is rolled. Find: ( 3 4 )P or , ( )P odd , (12 )P
7. Suppose a single card is selected from an ordinary deck of cards. Find:
a. (3 )P or k b. (3 )P H c. (3 )P J
8. The game cost $1 to play, and you get paid $10 if you draw a face, and you lose if any other outcomes. What is the expected value?
9. Which of the following is more probable? a. Correctly guessing all the answers on 20-questions true-false. b. Filliping coin 20-times and obtaining all heads.
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3.5 Expected Value
Objectives
Understand how expected values take both probabilities and winnings into account
Use expected values to analyze games of chance
Use expected values to make decisions
Expected Value
The expected value of an experiment is the average result you should expect to
approximate if you repeat the experiment a large number of time.
If an event E has several possible outcomes with probabilities p1, p2, p3, …, and if for
each of these outcomes the amount that can be won is a1, a2, a3, …, respectively, then the
mathematical expectation (or expected value) of E is:
Expectation = a1 p1 + a2 p2 + a3 p3 + …
Expectation = (Amount to win) (probability of winning)
Expectation with a cost of playing
If there is a cost of playing a game, the cost of playing must be subtracted from the
expectation. E = (Amount to win) P (win) – Expected cost
Ex:
10. Suppose you draw a card from a deck of 52 cards, and are paid $10 if it is king. What is the expected value?
11. Which game you choose to play?
a. Two dice are rolled. You will be paid $5 if you roll two fives. b. You will be paid $25 if you roll any pairs, but for any other outcome you pay $5.
12. A box contains one each of $1, $5, $10, and $20 bills. It cost $5 to reach in and withdraw one bill. What is the expected value?
13. A game of drawing a card from a deck of 52 cards if it is a face card, you win $20. If not you pay $ 5. Should you play the game? What is the cost to play to be a fair game?
14. A-50-questions multiple choices have 4-possible answers with one correct in each case, if someone guesses the answers randomly. What is the expected number of correct
answers?
15. Contest with first prize $1000, two second prizes $500 each, ten third prizes with $100 each, there was 0ne million entries. What is the expected value?
16. Suppose that you toss a coin and will win $1 if it comes up heads. If it comes up tails, you toss again. This time you will receive $2 if it comes up heads. If it comes up tails,
toss again. This time you will receive $4 if it is heads. What is the expectation for this
game?
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17. Suppose a family has three children. What is the expected number of boys? The sample space for this problem can be found using a tree diagram:
Number of Girls Probability Product
0 1/8 0
1 3/8 3/8
2 3/8 6/8
3 1/8 3/8
3 6 3 1
0 1.5 3 ( ) 3 1.5 8 8 8 2
E or E P G
18. If the probability of an event is 0.3 . What are the odd in favor for the event?
19. Flipping a coin and rolling a die. a. Find the odds in favor of 3. b. Find the odds against Tail.
20. Suppose a family wants to have 3-children. a. What is the sample space? b. What is: (2 )P G ?
c. What is: ( 2 )P at least G ?
d. Find the odds in favor of 2- G. e. Find the odds against 3- B.
21. Find the odds in favor of drawing a king from a deck if one card is drawn.
22. If the odd in favor of some event are 2 to 7. What is the probability of the event?
23. A game consists of rolling a die and receiving $12 if a one is rolled and nothing otherwise. What is the mathematical expectation?
24. What is a three-child family’s probability of having exactly two boys, given that at least one of their children is a boy?
25. What is the prob.(4-King)?
26. A single card is drawn from a stander deck of cards. What is the probability that it is a 3 if you know one 3 has been removed from the deck?
27. A box contains three orange balls and two purple balls, and two are selected at random. What is the probability of obtaining two orange balls if we know that the first draw was
orange and we draw the second ball:
a. After replacing the first ball? b. Without replacing the first ball?
28. There are ten seats, 6-red, and 4-green. If 3-people are assigned seats at random, what is the probability that they all receive green seats?
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3.6 Conditional Probability
Objectives
Use conditional probabilities to focus on one or two groups rather than the entire sample space.
Understand the relationship between intersections of events and products of probabilities
Use tree diagrams to combine probabilities.
Conditional probability
A conditional probability is a probability whose sample space has been limited to only
those outcomes that fulfill a certain condition.
The conditional probability of event A, given B, is: ( )
P (A / B) ( )
n A B
n B
Product Rule
The probability that both A and B happen, is: ( ) P (A / B) ( )P A B P B
( ) ( ) P (A ) ( )if A B P A B P B
Ex:
29. For a roll of a pair of dice, find the following. a. (3 )P on at least one of the dice
b. (3 4 )P on one die or on the other
c. (3 4 )P on one die and on the other
30. Suppose you draw a card from a deck of cards. Find a. P (face/jack) b. P (jack/face) c. P (heart/not a spade)
31. Suppose a single die is rolled. Find a. P (odd/the rolled number was a 5) b. P (4/even number was rolled)
32. Suppose a pair of dice is rolled. Find a. P (8/double was rolled)
P (a double/8 was rolled)
b. P (7/at least one die is 2)
33. How many plates can you create from (meat, chicken, fish), (mash potato, backed potato, French fries), (peens, Spanish, squash), (salad, soup).
34. Suppose a pair of dice is rolled. Find: (8/double was rolled)
35. What is the probability of having 5 girls in a row? What are the odds?
36. How many three letters words can be formed? ( repetition is allowed)
