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Transcript
Introduction to Probability
Learning Objectives
By the end of this lecture, you should be able to:
– Define the term sample space and event space. Be able to
identify those spaces through careful reading of a probability
question.
– Create and interpret a probability model.
“Yeah, yeah”
• Often we look at topics that seem like “common sense” and say to
ourselves, ‘yeah, yeah, I get that’. BE CAREFUL! Often there are many
ways in which we think we understand something, but there still remain
(many!!) gaps in our knowledge and understanding.
– This is not only true of statistics – happens in all kinds of places/studies.
– However, it is a particularly common pitfall in stats.
• Be sure to review the concepts and complement them with lots of
problems.
– There are not shortcuts – if there were I would tell them to you.
– Review and lots of problems is the only way to learn this material.
Randomness and probability
A phenomenon is random if individual
outcomes are uncertain. And yet, if the
probabilities are available, then given a
large number of repetitions, you can
expect to see a regular distribution of
outcomes.
For example, a single individual flip of a
coin is random. However a large number
of coin-flips will result in about 50%
heads and 50% tails.
The Concept: The “expected outcome” of a random phenomenon only applies
to a very long series of repetitions. For example, the probability of flipping a
heads is 50% given a large number of repetitions (flips).
Examples
• If you flip a coin, how often would you expect to get a heads?
– Answer: 50% of the time.
• If you flip a coin 10 times, can you safely expect to get 5 heads and 5 tails?
– Answer: No! In fact, try it… The reason is that 10 flips of a coin is a relatively low
number. You can only expect to get values close to 50% heads and 50% tails if you flip
the coin many times.
• If you roll a die, how often would you expect to get a 4?
– Answer: 1 in 6 times.
• Suppose you roll the die 6 times, can you safely assume that you will see a
4 one and only one time?
– Of course not! You can only expect to see the 4 show up exactly 1/6 times over many
rolls of the die.
Terms / Models – Learn ‘em!!!
• Sometimes we can intuitively figure things out.
– What is the probability of drawing an Ace of Spades from a deck of cards?
Answer: 1/52
– What is the probability of rolling a die and getting a 6 if on the previous roll
you got a 1? Answer: 1/6
• Sometimes, however, things become much more subtle and/or
complicated. This is why it is important to become very familiar with terms
and the concepts they refer to. In particular:
–
–
–
–
probability model
independent vs non-independent events
disjoint vs non-disjoint events
conditional probability
• Always focus on applying the proper concept/model and you will have a
much better chance at understanding the concepts and coming up with
the correct answers. Remember, it’s the concept that matters. If you don’t
have the concept, you risk incorrect statistics.
Sample Space
The sample space refers to the list of all possible outcomes of an event.
Determining the sample space sometimes requires careful reading of the question.
Key Point: Make sure that you correctly identify the sample space and have it clear in your mind. If
you make a mistake in identifying your sample space, you will end up with an incorrect solution to
the question.
– What is the sample space for the number of heads in three flips of the coin?
– S(# Heads) = {0, 1, 2, 3}
– What is the sample space for the number of 6s in two rolls of a die?
– S(# 6s) = {0, 1, 2}
– What is the sample space for the number of SuperBowls that could be won by the
Chicago Bears in the next 10 years?
–
– S(# SuperBowl Wins) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
What is the sample space for a randomly selected day of the week?
– S(Random Day of the Week) = {Sun, Mon, Tue, Wed, Thu, Fri, Sat}
Note: The best way to make sure you have correctly identified the sample space, is to make sure that
you read the question carefully!
Sample Space
It’s the question being asked that determines the sample space.
A. A basketball player shoots
three free throws. What are
the possible sequences of
hits (H) and misses (M)?
H
H -
HHH
M -
HHM
H
M
M…
H -
HMH
M -
HMM
S = { HHH, HHM,
HMH, HMM, MHH,
MHM, MMH, MMM }
…
B. A basketball player shoots
three free throwsWhat is the
number of baskets made?
S = { 0, 1, 2, 3 }
C. A nutrition researcher feeds a new diet to a young male white rat. What
are the possible outcomes of weight gain (in grams)?
S = [0, ∞[
(the ‘[‘ means that infinity is excluded)
Probability of Outcomes in the Sample Space
•
Every outcome in a sample space has some probability of occuring.
– We notate this outcome as P(….)
– Eg: Looking at a single coin flip. Let X = the flip result. What is P(X=heads)?
• Answer: P(X=Heads) = 0.5
– Eg: Looking at two coin flips, Let X = the number of heads. What is P(X=2)? Answer is 0.25 – we will
explain how we got this number later.
– For the moment, an important point here, is to note the terminology/symbology being used.
Each one of the possible outcomes of a sample space may have different probabilities.
• Outcomes of 1 flip of a coin:
– S = {Heads, Tails}.
• In this case, each event in the sample space has the same probability of occurring.
– P(Heads) = 0.5, P(Tails) = 0.5
• An at-bat baseball player’s outcomes for getting a hit or getting an out has the sample space S =
{hit, out}. Note that each outcome in this sample space has a different probability. For example:
– P(Hit) = 0.294, P(Out) = 0.706
• A good professional basketball player may hit 70% of free throws. S = {hit, miss}
– P(Basket) = 0.7, P(Miss) = 0.3
Outcomes / Events
•
It is important to be able to CLEARLY identify the sample space and also the
event(s) you are interested in. If you do not read the question with some care, you
can easily mis-identify one or both of these.
– Without proper identification of the sample space and the event, you’re gonna
end up with the wrong answer!
•
An event is a list of the particular outcomes that you are interested in.
•
An event is typically some subset of the sample space. Notice that the event is NOT
the same thing as the probability. The event is simply the result you are interested in
calculating a probability for. The probability refers to the likelihood of that event
occuring.
Example: Identify the Outcomes and Events
REVIEW:
• The sample space is a list of all possible outcomes.
• An event is a list of any particular outcomes that you are interested in.
•
For the following two examples, define the sample space and the event. Then rephrase the
probability question using appropriate symbols:
– If I roll a single die what is the probability that I will roll a 1?
• Sample space = {1, 2, 3, 4, 5, 6}
• Event = {1}
• Let X = the outcome of the roll. What is P(X=1)?
– If I roll two die, what is the probability the sum will be a 7 or an 11?
• Sample space = {2,3,4,5,6,7,8,9,10,11,12}
• Event = {7, 11}
• Let X = the sum of the two die. What is P(X=7 or X=11)?
An event is always some subset of the sample space. Notice that the event is NOT the same thing as
the probability. The event is simply the result you are interested in calculating a probability for. The
probability refers to the likelihood of that event occuring.
Probability Models
Probability models describe, mathematically, the outcome of random
processes. A probability model consists of two parts:
1) Sample Space (‘S’): This is the set, of all possible outcomes of
a random process.
2) A probability for each of the possible outcomes in the sample
space.
Example: Probability Model for a Coin Toss
Sample Space:
Heads
Tails
Probability:
0.5
0.5
Probability Model for a Die Roll
Die Roll
(Sample Space)
1
2
3
4
5
6
Probability:
1/6
1/6
1/6
1/6
1/6
1/6
Probability Model Example
Example: A couple wants three children. What are the numbers of girls they could end up
with? What are the probabilities for each outcome? Create a probability model.
Also: Be sure to note and be comfortable with the way we use variables and symbols such as P(X=2)
Sample space
S = {0, 1, 2, 3}
 P(X = 0) = P(BBB) = 1/8
 P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) = 3/8
 P(X = 2) = P(BGG or GBG or GGB) = P(BBG) + P(BGB) + P(GBB) = 3/8
 P(X = 3) = P(GGG) = 1/8
Probability Model:
Terminology Example: How many girls might the couple have?
•
•
•
We will use the probability model from the previous example.
It is vitally important to clearly define what X represents. In this case, X is a count
of the number of girls that the couple will end up with out of their 3 children.
So if your statistics prof gives you the situation in the previous slide and then
writes P(X=2) what exactly are you being asked?
– Answer: “What is the probability of the couple ending up with exactly two girls?”
•
If you were asked P(X>2)?
– Answer: What is the probability of the couple ending up with exactly 3 girls?
•
What if you were asked P(X>=2)?
– Answer: What is the probability that the couple ending up with either 2 girls or 3 girls?
•
You must be very clear with this concept. Note that on this slide, I do not spend the
time working out the mathematical answer. This is intended as a reminder to you
that the most important thing to do first is to clearly understand what the question
is asking and to understand precisely what the variables represent.
Coin Toss Example:
S = {Head, Tail}
Probability of heads = 0.5
Probability of tails = 0.5
Probability rules
1) Probabilities range from 0
(no chance of the event) to
1 (the event will always happen).
For any event A, 0 ≤ P(A) ≤ 1
Probability of getting a Head = 0.5
We write this as: P(Head) = 0.5
P(neither Head nor Tail) = 0
P(getting either a Head or a Tail) = 1
2) The sum of the probabilities
of all the possible outcomes (the sample
space) must be exactly 1.
P(sample space) = 1
Coin toss: S = {Head, Tail}
P(head) + P(tail) = 0.5 + 0.5 =1
 P(sample space) = 1
Example: A basketball free-thrower who makes the
basket 72% of the time:
– S = {Hit, Miss}
– P(Hit) = 0.72
– P(Miss) = 0.28
– P(Hit + Miss) = 1.0
• Recall, all probababilities in the sample space must add up to 1.
Key Point: Note that the sum of all probabilities must add up to 1. In this
case, the two outcomes, Hit & Miss add up to 1.