Name: ______
Chapter 6: Probabilityand Simulation
trialrandom probability independence
random phenomenonsample space S = {H, T} tree diagram
replacementevent P(A) Complement AC
disjointVenn Diagram union (or) intersection (and)
joint eventjoint probability conditional probability
Introduction – Answer the following questions
1a. What is likelihood of getting 3+ consecutive heads or tails if you toss a coin 10 times______
1b. You are going to have children until you have a girl or have 4 children whichever comes first. What are the chances that you will have a girl? ______
1c. If an airline overbooks a flight what are the chances that the airline will encounter more passengers than they have seats for? ______
2. List 3 methods to answer these questions
1.
2.
3.
6.1 Simulation
1. Read ex 6.1 Three scenarios where simulation is helpful. Can you think of another example when a simulation would be helpful? ______
2. Name 2 simple simulations for 1b above? ______, ______
3. Define simulation -
4. Simulation steps
1.
2.
3.
4.
5.
5. Define trials –
6. Read ex 6.4 a simulation when outcomes are not equally likely
7. Read ex 6.5 simulation using random digit table
8. Read ex 6.6 girl or 4 children
9. Read ex 6.7 using the calculator
10. Do Activity 6B page 399
11. Do ex 8 on your calculator
12. Complete TPS 6.1, 6.3, 6.13
13. Complete TPS 6.3
14. Complete TPS 6.13
6.2 Probability Models
15. Activity 6C pg 406 - graphing calculator
16. Chance behavior is ______in the short run but has a ______and ______pattern in the long run.
17. Activity 6D pg 407 – Applet
18. Random in statistics does not mean ______. It means a ______
______.
19. empirical –
20. random – individual outcomes are ______but ______distribution of outcomes in a ______.
21. probability of any outcome - ______of times outcome would occur in
______. It is a ______.
22. Probability theory describes ______.
23. Random
* Must have long series of ______trials which means that outcome of one trial ______.
* probability is ______
* Computer simulations allow us to see the results of ______.
24. What started the study of probability?
25. What led to advancements in probability?
26. List a few ways we use probability today.
6.3 Probability Models
1. sample space –
Event –
Probability model –
Ex 10 outcomes of sample space
2. Draw a tree diagram for number of different meal combinations given the following
Choice of tea/coke, sandwich/pasta, salad/fries/fruit
3. How could you use the multiplication principle to solve this problem instead of doing a tree diagram?
4. You have 4 red marbles and 3 green marbles in a bag.
What is the probability of 1st draw – red, 2nd draw – green with replacement?
What is the probability of 1st draw – red, 2nd draw – green without replacement?
5. Probability Laws (write in words and mathematical symbols)
1.
2.
3.
4.
6. Draw a Venn Diagram for the following let A= even #, B = 3, C = 4. Calculate answer.
P(A or B) =
mutually exclusive (disjoint)
P(A or C) =
inclusive
7. What does the complement mean?
P(rolling a 3) = complement of P(rolling a 3)
8. Rolling pair of dice, what is P(rolling a 5) =
Probabilities in a finite sample space –
9. ex 15 - Benford’s Law, Clever crooks
10. problem 44
11. problem 38
12. P(A) and P(B) = P(A) x P(B) if A and B are ______which means ______.
13. ex 17 and 18 and 19 independence
14. Disjoint (mutually exclusive) events cannot be ______.
15. Why can’t independence be pictured in a Venn diagram?
16. 75% of the people in this neighborhood graduated from college. If we choose people at random, what is the probability that the 1st person is a college graduate and the 2nd person is not?
17. ex 21 aids testing false positives
18. problem 49
19. Rules of Probability
1.
2.
3.
4.
5.
20. P(A u B) = A or B or ______occur
21. uniform distribution – draw a density curve for rolling a number cube
What is the probability of rolling an odd number?
22. Draw Venn Diagram and calculate the probability. Using standard deck of 52 cards, -
P(King) = P(face card) =
23. P(A or B) = P(A) + P(B) – P(A and B)
P(A u B) = P(A) + P(B) – P(A n B)
24. like ex 23 probability addition rule, Sue /Bob promotions
Probability Sue makes partner is 0.5, Bob’s probability is 0.75
Probability that both are promoted to partner is 0.4
P(at least one is promoted) =
P(neither one is promoted) =
Draw a Venn diagram representing this problem with #’s
BobProm / Not / TOTAL
Sue / Prom.
Not
TOTAL
P(S and B) = P(Sue and Bob promoted) =
P(S and Mc) = P(Sue promoted and Bob not) =
P(Sc and M) = P(Sue not and Bob promoted) =
P(Sc and Mc) = P(Sue is not and Bob is not) =
These joint probabilities must all add up to ______.
25. Conditional probability –
26. What is the probability of drawing a Queen given that I know it’s a face card?
P(queen l face card) =
27. ex 25 grades in college
CAREFUL WITH PROBABILITIES AND CONDITIONAL PROBABILITIES
28. Go back to ex. 25 P(A) = grade from Liberal ArtsP(B) = grade is an A
P(A) = P(B) = P(A and B) = P(B l A) =
Now prove P(A and B) = P(A) x P(B l A)
so P(B l A) =
29. ex 26 and 27
Caution P(B l A) = P(A) > 0
30. P(A and B and C) =
Like ex 29 P(competes in college) = P(A) = .15
P(competes professionally) = P(B)P(B l A) = 0.07
P(competes pro > 3 years) = P(C) P(C l A and B) = .50
What is the probability that a player competes in college and then pro > 3 years?
31. like ex 30 with the following #’s
Age% of internet users% that chat
18-29.35.70
30 – 49.50.30
50+.15.35
Make a tree diagram
What is the probability that a randomly chosen Internet user participates in chat rooms?
Probability of reaching the end of any complete branch is the ______.
Probability of any outcome is found by ______that are part of the event.
32. What percent of chat room participants are 18-29 year old? Use #’s from 31 above
33. Two events are ______if P(B l A) = P(B).
If knowing that A occurs gives you ______then A and B are independent.
Chapter 6 Sec 6.1
Chance is all around us. Probability is the branch of mathematics that descries the pattern of chance outcomes. The reasoning of statistical inference rests on asking, "How often would this method give a correct answer if I used it very many times?" When we produce data by random sampling or randomized comparative experiments the laws of probability any that question. The fundamental concepts of probability are the basis for inference. The tools you acquire will help you describe the behavior of statistics from random samples and randomized comparative experiments later.
The mathematics of probability begins with the observed fact that some phenomena are random... that is, the relative frequencies of their outcomes seem to settle down to fixed values in the long run.Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.You cannot predict the outcome of tossing a coin one time but if you make many, many tosses a regular pattern will emerge....this is the REMARKABLE fact that forms the basis for probability. The proportion of tosses that produce heads is quite variable at first, but settles down as we make more and more tosses as shown in Ex. 6.1 on page 331.
Random in statistics is NOT a synonym for "haphazard" but a description of a kind of ORDER that emerges only in the long run. The idea of probability is empirical which means based on observation rather than theorizing. When presented with opportunity, some mathematicians tested the coin toss results.
A little bit of trivia...Count Buffon tossed a coin 4040 times resulting in 2048 heads or 2048/4040 = .5069. Karl Pearson tossed a coin 24,000 times resulting in 12,012 heads or 12,012/24,000 = .5005. while a POW during WWII John Kerrich tossed a coin 10,1000 times with 5067 heads or 5067/10,000 = .5067. These results are extraordinarily alike indicating the conclusion that the probability of tossing a head on a fair coin settles down to around 50%.
A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.
Probability is the branch of mathematics that describes random behavior. Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials. These trials must be INDEPENDENT, that is, the outcome of one trial must NOT influence the outcome of any other.
Some statistical history/evolution...Probability theory originated in the study of games of chance. Tossing dice, dealing shuffled cards, and spinning a roulette wheel are examples of deliberate randomization that are similar to random sampling. During the 17th century French gamblers asked mathematicians, Blaise Pascal and Pierre de Fermat for help. Gambling is still with us in casinos and state lotteries. In the 20th century the mathematics of probability is used to describe the flow of traffic through a highway system, a telephone interchange or a computer processor; the genetic makeup of individuals or populations, the energy states of subatomic particles; the spread of epidemics or rumors, and the rate of return on risky investments. Although we are interested in probability because of its usefulness in statistics, the mathematics of chance is important in many fields of study.
Let's do Ex. 6.3 on page 334 to practice these basic concepts.
Chapter 6 Sec 6.2(a)
We have already learned how to used models for linear relationships and for some normal distributions in the normal density curve. Now we create a mathematical description or model for randomness. First think about a very simple random phenomenon, tossing a coin once. We describe the coin toss in two parts: a list of possible outcomes and a probability for each outcome. The list of all the possible outcomes is called the SAMPLE SPACE (S). An event is any outcome or set of outcomes of a random phenomenon. An event must be present in the sample space. A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space (S) and a way of assigning probabilities to events.
To specify S we must state what constitutes an individual outcome and then state which outcomes can occur. We have some freedom in defining the sample space and make choices through convenience. Being able to properly enumerate the outcomes in a sample space will be CRITICAL to determining probabilities. Creating a "tree diagram", as used in elementary school, can be helpful with all of its branches labeled. Learning the MULTIPLICATION PRINCIPLE for events occurring is also helpful. The rule states:
If you can do one task in "x" number of ways and a second task in "y" number of ways, then BOTH tasks can be done in x times y number of ways. Try this yourself before seeing the explanation in the book.
An experiment consists of flipping four coins. You can think of either tossing four coins on the table at at once OR flipping a coin four times in succession and recording the four outcomes. One possible outcome is HHTH. Because there are two ways each coin can come up, the multiplication principle says that the total number of outcomes is2 X 2 X 2 X 2 =16. See if you can list all the possibilities...try to describe your method.
It is not a huge step from tossing coins to polling four people at random regarding whether they favor reducing federal spending on low-interest student loans. Each has only two possible outcomes. Even when the numbers are increased to 1500 or some such for the poll, it is analogous to 1500 coin tosses. ONE OF THE GREAT ADVANTAGES OF MATHEMATICS IS THAT THE ESSENTIAL FEATURES OF QUITE DIFFERENT PHENOMENA CAN BE DESCRIBED BY THE SAME MATHEMATICAL MODEL. Read the last sentence again and again until you grasp its importance. (Remember all the different types of word problems that resulted in linear models, or quadratic models, or exponential modes from Algebra 2???)
In the case of coin tossing or dice rolling, the sample space has a finite number of outcomes possible. Sometimes the sample space is infinite as when a computing system has a function that generates a random number between 0 and 1. S is a mathematical idealization and in this case would best be described using an interval.
IF you select random digits by drawing numbered slips of paper from a hat, and you want all ten digits to be equally likely to be selected each draw, then after you draw a digit and record it, you MUST put it back into the hat...then the second draw will be exactly like the first. This is called sampling WITH REPLACEMENT. IF you do NOT replace the slips you draw, there are only nine choices for the second slip picked, and eight for the third and so on. This is called sampling WITHOUT REPLACEMENT. So...if the question is "How many
3-digit numbers can you make?" the answer is 10 X 10 X 10 = 1,000 ways by the multiplication principle PROVIDING all ten numbers are eligible for each of the three positions in the number. On the other hand, there are 10 X 9 X 8 = 720 different ways to construct a three-digit without replacement. You should be able to determine from the context of the problem whether the selection is with or without replacement and this will help you identify the sample space.
Probability has specific rules and notation that will guide us with more complex situations:
1. any probability is a number between 0 and 1. An event with probability = 0 NEVER occurs, and an event with probability 1 occurs on every trial. Symbolically, 0 P(A) 1 where A is an event and P(A) is the probability of that event occurring.
2. All possible outcomes together must have probability 1...the sum of the probabilities for all possible outcomes must be exactly 1. All possible outcomes is another way to say sample space, so symbolically P(S) = 1.
3. The probability that an event does NOT occur is 1 minus the probability that the event DOES occur. The probability that an event OCCURS and the probability that is does NOT occur always add to 100%, or 1.
Symbolically, P(Ac) = 1 - P(A) where Ac refers to the complement of A which is A does not occur.
4. If two events have no outcomes in common, the probability that one OR the other occurs is the SUM of their individual probabilities. Symbolically, P(A or B) = P(A) + P(B) iff the events are disjoint or mutually exclusive. Disjoint means have nothing in common.
The use of Venn diagrams and the use of set notation is very helpful for picturing and talking about events occurring. The event {A U B} read "A union B", is the set of all outcomes that are EITHER IN A OR IN B. This is another way to say "OR" like rule #4 above. The symbol 0 is used for empty event. If two events A and B are disjoint, we write "A intersect B" is empty. Sometimes we emphasize that we are describing a compound event by enclosing it within braces. In Venn diagrams there will be NO overlap of the A and B event areas. Some other logical conclusions are "A union Ac" = S and "A intersect Ac" = empty set.
Chapter 6 Sec 6.2(b)
Some things to remember when assigning a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum 1.The probability of any event is the sum of the probabilities of the outcomes making up the event.
In some circumstances we assume that individual outcomes are equally likely like ordinary coins having a physical balance that should make heads and tails equally likely. The table of random digits comes from a deliberate randomization making each number equally likely.
There is a special rule for determining the probability of event A from a group of equally likely outcomes is:
P(A) = outcomes in A / outcomes in S OR in English desired outcomes / total outcomes
Note: Most random phenomena do NOT have equally likely outcomes, so the general rule for finite sample spaces is more important than the special rule for equally likely outcomes.
Rule 4 (addition rule for disjoint events) describes the probability that one OR the other of A and B will occur when A and B cannot occur together. Next we describe the probability that BOTH events A and B occur in another special situation.
Independent events: two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P( A and B) = P(A)P(B). This is called the MULTIPLICATION RULE FOR INDEPENDENT EVENTS.Independence is usually assumed as part of a probability model when we want to describe random phenomena that seem to be physically unrelated to each other. The multiplication rule extends to MORE than two events provided that all are independent.
Repeating...the multiplication rule P(A and B) = P(A)P(B) holds only if A and B are INDEPENDENT and not otherwise. The addition rule P(A or B) = P(A) + P(B) holds if A and B are disjoint but not otherwise. So.....
disjoint events are not independent. A Venn diagram can indicated disjoint but does not indicate independence.
If two events A and B are independent, then their complements Ac and Bc are also independent and Ac is independent of B. By combining the rules we have learned that we can compute probabilities for complex events. See Ex. 6.15 on page 354.
Probability thus far...
a) a random phenomenon has outcomes that we cannot predict but that do have a regular distribution
in many repetitions
b) the probability of an event is the proportion of times the event occurs in many repeated trials
c) a probability model consists of a sample space S and an assignment of probabilities P
d) the sample space S is the set of all possible outcomes
e) sets of outcomes are events
f) P assigns a number P(A) to an event A
g) the complement of Ac consists of exactly the outcomes that are NOT in A
h) A and B are disjoint (mutually exclusive) IF they have no outcomes in common
i) A and B are independent if knowing one event occurs does not change the probability of the other.
Five Rules so far...
1) All probabilities lie between 0 and 1.
2) P(S) = 1 (all outcomes added together have probability of 1)
3) COMPLEMENT RULE: for any event A, P(Ac) = 1 - P(A) from rule 2 and defn of complement
4) ADDITION RULE: if A and B are DISJOINT then P(A or B) = P(A) + P(B). (Union of events)
If A and B are disjoint A AND B can NEVER occur together.
5) MULTIPLICATION RULE: if A and B are INDEPENDENT then (A and B) = P(A)P(B)
(interesection of events).
Statistics means never having to say you're certain.
Chapter 6 Sec 6.3
The purpose of learning more laws of probability is to be able to give probability models for more complex random phenomena. We already have five rules from the last section. Our addition rule only applies to DISJOINT events. So...what happens when events "overlap" some???
GENERAL RULE FOR UNION - (OR) for two events:
For ANY two events A and B, P(A or B) = P(A) + P(B) - P(A and B)
Note: When A and B are disjoint P(A and B) = 0 and the rule reverts to the addition rule of before
Venn diagrams are a help in finding probabilities for unions, because you can just think of adding and subtracting areas.
The simultaneous occurrence of two events A and B is called a JOINT event. The probability of a joint event is called a JOINT PROBABILITY. Working with joint events is best done with a table as described on pages 363 and 364.