Probability
Probability
Probability
Definition of Probability: A value between ____ and _____ , inclusive, describing the
relative possibility (chance or likelihood) an event will occur.
Inferential decisions are based on probabilities or likelihoods.
Complementary Events, A
What is the probability a randomly selected doctor from the hospital is:
• Subjective Approach…
We define probability as the degree of belief that we hold in the occurrence of an event”
Addition Rule: P(A or B) = P(event A occurs or event B occurs or they both occur)
Independent Events: Two events A and B are independent if the occurrence of one
________ the probability of the occurrence of the other.
If A and B are not independent, then they are called dependent.
Multiplication Rule:
When drawing two cards from a shuffled deck, find the probability that
the first card is an ace and second card is a king.
(a) Assume that the first card is replaced before the second card is drawn.
(b) Assume that the first card is not replaced before the second card is drawn.
Extra: What is probability that the first card is an ace and second card is an ace if the first
card is not replaced?
Reference:
Conditional Probability: The probability of a particular event occurring, given that another
event has occurred.
Notation: P(B|A) represents the probability of B occurring after it is assumed that the
event A has already occurred. (read B| A as “the probability of B given A”)
Quick Notes
P( A | B) =
P( A & B) Conditional
P( B) probability formula
Only one event The subtotal of a column or row/ Total Marginal Probability
The following table shows the results of a survey in which 186 smoking and 414
non-smoking men ages 60 to 65 were asked if they had lung disease.
(a) He is a smoker.
(f) If he is a nonsmoker, what is the probability that he does not have lung disease?
P ( A and B)
P ( B | A) =
P ( A)
7 Prepared by Prof. Kobayashi
STAT 2263 Probability
The following table shows a fictitious statistical study of a new acne cream.
(b) Find the probability that someone used a placebo and skin improved.
(c) Find the probability that someone used a placebo or someone’s skin did not improve.
(d) The probability that someone’s skin improved, given that they used the new acne
cream is:
P ( A and B)
P ( B | A) =
P ( A)
8 Prepared by Prof. Kobayashi
STAT 2263 Probability
Tree Method
A probability tree is a simple and effective method of applying the probability rules by
representing events in an experiment by lines.
Bayes’ Theorem
A manufacturer claims that its drug test will detect steroid use (that is, show positive for an
athlete who uses steroids) 95% of the time. What the company does not tell you is that 15% of
all steroid-free individuals also test positive (the false positive rate). 10% of the rugby team
members use steroids.
Tree
95 % sensitivity =True Positive: P(+ S ) (positive result given that used steroid) =0.95
False Negative: P(− S ) (negative result given that used steroid) = 1 – 0.95 =0.05
85% specificity = True Negative: P ( − S ) (negative result given that did not use steroid) = 0.85
False Positive: P ( + S ) (positive result given that did not use steroid) = 1 – 0.85 = 0.15
(a) Find the probability that the member did not use steroids and the test result is positive.
(c) What is the probability that the member did not use steroids given that the test result is negative?
P ( A and B)
P( A | B) =
P( B)
10 Prepared by Prof. Kobayashi
STAT 2263 Probability
If a woman takes an early pregnancy test, she will either test positive, meaning that the test
says she is pregnant, or test negative, meaning that the test says she is not pregnant.
Suppose that if a woman is really pregnant, there is a 97% chance that she will test positive.
Also, suppose that if a woman really is not pregnant, there is a 98% chance that she will test
negative. Suppose that 5% of women who take early pregnancy test really are pregnant.
Tree
(a) What is the probability that a randomly chosen woman from this group is pregnant given that
she tests negative?
P ( A and B)
P( A | B) =
P( B)
(b) If a woman tests positive, what is the probability that a randomly chosen woman from this group
is not pregnant?
P ( A and B)
P( A | B) =
P( B)
11 Prepared by Prof. Kobayashi