Download 4.1 Prob dist and expected value

Unit 4: Probability Distributions
and Predictions
4.1 Probability Distributions and
Expected Value
Posing and
refining questions
Collecting data and
finding information
Making sense
of the data
Making the case
Evaluating the
conclusions
A certain section of a course has a 4%
failure rate. Is that good or bad?
The results of a clinical trial of a certain
test of a disease show a 15% false
positive rate. If you are tested positive
for the disease, should you followed the
prescribed medical treatment?
Probability Distributions
• The last unit looked at probability of
individual outcomes of an experiment
• Now, we will look at the distribution of
probabilities of all possible outcomes
• Distributions can involve outcomes with
equal or uneven probabilities
Experiment
• You will receive 2 dice
• Roll the two dice and record the sum n
times
• We will combine the class results
• Calculate the experimental probability for
each sum
• Create a probability distribution table of the
possible sums
Sum of Two Dice
X
2 3 4
5
6
7
8
9
10 11 12
P(X=x)
• Experimental Probability Distribution
• Theoretical Probability Distribution
X
2 3 4
5
6
7
8
9
10 11 12
P(X=x)
1 2 3 4
36 36 36 36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
Probability Distributions
• Provides probability of each possible value of
random variable X
• Can be given in table or graph form
Histogram
Sum of 2 dice
0.35
Relative Frequency
16
14
12
Count
Histogram
Sum of 2 dice
10
8
6
4
0.30
0.25
0.20
0.15
0.10
0.05
2
0
2
4
6
8
sum
10
12
14
0
2
4
6
8 10
sum
12
14
• Relative frequency distribution shows ratio of
frequency to total number of trials
16
Variables
• Many probability experiments have
numerical outcomes
• Random variable, X
– Single value (denoted x) for each outcome
• Discrete variables
– Values are counted
• Continuous variables
– Infinite number of possible values on a
continuous interval
Are the following variables
discrete or continuous?
•
•
•
•
•
•
Number of times you catch a ball
Length of time you play ball
Length of car in centimetres
Number of red cars on highway
Volume of water in a tank
Number of candies in a box
discrete
continuous
continuous
discrete
continuous
discrete
We will be looking at discrete random variables for now.
Uniform Probability Distribution
• All outcomes in distribution equally likely
in any single trial
• Probability of discrete uniform distribution
1
P( X  x) 
n
for all possible x
Expected Value, E(X)
• The value you would expect to get for one
trial of an experiment
– E.g.: The theoretical probability for sum of two
dice is highest for 7
– You would expect to get a sum of 7
• E(X) is the predicted average of all possible
outcomes
E ( X )  x1P( X  x1 )  x2 P( X  x2 ) 
 xn P( X  xn )
It’s actually just a weighted mean (the probabilities are the weights)!
E(X) of Sum of Two Dice
E ( X )  x1P( X  x1 )  x2 P( X  x2 ) 
n
 xn P( X  xn )
  xi P ( X  xi )
i 1
 2  P( X  2)  3  P( X  3) 
11 P( X  11)  12  P( X  12)
1
2
3
4
5
6
 2   3  4   5  6   7 
36
36
36
36
36
36
7
5
4
3
2
1
8   9   10   11  12 
36
36
36
36
36
So?
• Why would we want to know the expected
value of an experiment?
• We can compare the expected value to
actual values
• Are the actual values “reasonable?”
– That is, is that what we could expect to get?
Example 2
You pull 3 books out of your locker at
random. You have 3 math/science books
and 4 English/history books.
a) What is the probability that at least two of
the books are for a math/science class?
b) Create a probability distribution table for
the possible book selection.
c) How many math/science books could you
expect to pull out of your locker?
Example 2a sol’n
n( A)
P( A) 
n( S )
 3   4   3  4 
      
2   1   3  0 


7
 
 3
 0.371
Therefore there is a
37.1% chance of pulling
out at least two
math/science books.
Example 2b sol’n
Combina- 0 math/sci 1 math/sci 2 math/sci 3 math/sci
tions
3 Eng/hist 2 Eng/hist 1 Eng/hist 0 Eng/hist
Proba0.114
0.514
0.029
bility
0.343
Calculations:
What is the sum of all the probabilities?
1
Example 2c sol’n
What is the discrete random variable X?
X is the number of math/science books pulled out of
the locker.
4
E ( X )   xi P( X  xi )
i 1
 0  P( X  0)  1 P( X  1)  2  P( X  2)  3  P( X  3)
 0(0.114)  1(0.514)  2(0.343)  3(0.029)
 1.3
You can expect to pull out about
1.3 math/science books.
What is the expected value of a fair game?
0
Why?
The probabilities should be evenly distributed and
the wins should balance the losses
No game in a casino is fair.
The expected value on a simple bet of a game of
roulette is -$0.053.
That is, for every spin, the casino expects to make
about 5 cents.
(How many spins are there in an average year?)
Roulette Odds
Result of spin
Win (ball
lands on your
number 1-36)
Lose (ball
lands on other
number or 0 or
00)
x
P(X = x)
$35
1
38
-$1
37
38