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Introduction to Probability
Probabilities are expressed in terms of events
Examples of Events:
- Six shows on a roll of a die
- Jack of Spades is drawn from a deck of cards
- Rains at 3:00 p.m. today in front of the Reitz Union
- Have an automobile accident this year
- Florida beat Florida State
- Florida beat Tennessee
- Florida beat Florida State and Tennessee
- Yield of a randomly drawn citrus tree is greater than 8 boxes
- Mean yield of 25 randomly drawn citrus trees is greater than 8 boxes
Events are denoted by capital letters, A, B, etc.
Probabilities of events are denoted P(A), or P(event occurs)
Determination of Probabilities
Probabilities are determined from:
- Relative frequency computation
- Subjective assessment
P(six on roll of die)=1/6, a relative frequency computation
P(Florida beat Florida State)=.3, a subjective assessment
P(Rain today at 3:00 at JWRU)=.4, combination of relative frequency and subjective assessment
Calculating Probabilities of Compound Events
P(AUB)=P(A)+P(B)-P(A ∩ B)
P{Florida beat either FSU or Tennessee}=P{Florida beat FSU}+P{Florida beat Tennessee}-P{Florida beat both FSU and Tennessee}
Two events, A and B, are independent if P(A ∩ B)=P(A)P(B)
Are the events {Florida beat FSU} and {Florida beat Tennessee} independent?
Are the events {Drives 4WD truck} and {Voted Republican} independent?
Are the events {Lives in Florida} and {Voted Republican} independent?
Conditional Probability
The conditional probability of A given B is the probability that A occurs, when it is known that B occurs. It is calculated by the formula
P(A|B)=P(A∩B)/P(B).
If A and B are independent, then P(A)=P(A|B)
Application
50 1 P(PC+) =. 10000 200 398 P(PSA+) =. 10000 P(PC+ PSA+) = 48. 10000
= =
=
∩ =
Conditional Probability: P(A given B) = P(A|B) =
P(A B) P(B)
∩
P(PSA+ |PC+ ) = 48/50 = .96 = sensitivity
P(PSA- | PC-) = 9600/9950 = .9648 = specificity
P(PC+ |PSA+) = 48/398 = .12 = predictive ability
Prostate Cancer
- (^48 350 ) PSA Test -^2 9600
50 9950 10000
Random Variables:
Discrete
spots on top face of die (1, 2, 3, 4, 5, 6)
- suit of drawn card (C, D, H, S)
aphids on leaf (0, 1, 2, 3, …)
defects in box of 1000 nails (0, 1, 2, …, 1000)
germinating seeds out of 50 (0, 1, 2, …, 50)
Continuous
- heights of people (0 -- ?)
- ph of soil (0 – 10)
- voltage in circuit (0 -- ?)
Binomial Random Variable (con’t)
Example: x = number of 1’s in 3 rolls of die (0, 1, 2, 3)
Events: 111 11X 1X1 X11 1XX X1X XX1 XXX
1’s 3 2 2 2 1 1 1 0
P(Event) 1 3 /6 3 1 25 1 /6 3 1 25 1 /6 3 1 25 1 /6 3 1 1 · 5 2 /6 3 1 1 · 5 2 /6 3 1 1 · 5 2 /6 3 5 3 /
P(3 1’s) =
3
P(k 1’s) =
3
k k
k k
−
Binomial Formula
π = probability of success on single trial
P(y successes in n trials) =
n y n y
y n y
Mean of the binomial distribution: n^ π
Variance of the binomial distribution: n^ π^ (1^ −π)
0.
0.
**0.
0 1 2 3**
Normal Distribution and normal random variable :
The notation y ~ N(μ,σ^2 ) means “The random variable y is distributed normally with mean μ and variance σ^2.
The standard normal distribution has mean μ=0 and variance σ^2 =1. The letter z is reserved to represent the standard normal random variable.
Computer programs and tables are available to obtain probabilities from the normal distribution. For example, you can discover that
- P(-1 < z < 1) =.
- P( z > 1) =.
- P(-1.96 < z < 1.96) =.
- P( z > 1.96) =.
- P( z > 1.42) = .0778.
Using the normal distribution—an application
Egg weights are normally distributed with mean μ^ =^65 (g) and
standard deviation σ^ =^ 5.0.
- What is the probability one randomly drawn egg will exceed: a. 65 b. 66 c. 70 d. 75
Let y = egg weight. Then a. P(y > 65) = 65 65 65 5 5 P ⎛^^ y −^ > − ⎞ ⎜⎝ ⎟⎠ = P(z > 0) = ½ =.
b. P(y > 66) = 65 66 65 5 5 P ⎛^^ y −^ > − ⎞ ⎜⎝ ⎟⎠ = P(z > .2) =.
c. P(y > 70) = 65 70 65 5 5 P ⎛^^ y −^ > − ⎞ ⎜⎝ ⎟⎠ = P(z > 1) =.
d. P(y > 75) = 65 75 65 5 5 P ⎛^^ y −^ > − ⎞ ⎜⎝ ⎟⎠ = P(z > 2) =.
- What is the probability one egg is between 66 and 70 g? P(66 < y < 70) = P(y > 66) – P(y > 70) = .4207 - .1587 =.
Normal Approximation to the Bionomial
You can use the normal distribution to approximate binomial probabilities. This often simplifies a computation. For example, suppose you are shooting free-throws in basketball. You know that you make 75% of your shots; that is, the probability of making any one shot is .75. You have entered a competition that awards a prize if you make at least 18 out of 20 shots. What is the probability that you will win a prize?
You need to calculate P( y ≥18), where y is the number of shots you make out of 20. The exact probability is given by the binomial formula with π = .75 and n = 20:
P( y ≥ 18) = P( y = 18) + P( y = 19) + P( y = 20)
= 20!/(18!2!).75 18 .25^2
+20!/(19!1!).75 19 .25^1
+20!/(20!0!).75 20 .25^0
