What is probability Classical definition the ratio of favorable to equally probable cases favorable the kind youre interested in Probability of getting heads on flipping a fair coin 12 heads is 1 of 2 possibilities ID: 779352
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Slide1
Introduction to Probability
Slide2What is probability?
Classical definition:
the
ratio
of “favorable” to equally probable
cases
.
“
favorable”
:
the kind you’re interested
in
.
Probability of getting heads on flipping a fair coin: 1/2 (heads is 1 of 2 possibilities)
Slide3What is the probability of
getting heads twice on two tosses?Four possibilities:
Case 1
Case 2
Case 3
Case 4HHHTTHTT
one
out of four (1/4): just Case 1
What
is the probability of
getting heads at least once on two tosses
?
3/4: Cases 1, 2, and 3
Slide4Communicating and
r
easoning
a
bout
probabilities25%25 out of 1001/4.25
Slide5Communicating and
r
easoning
a
bout
probabilities“If you take Prozac you have a 30–50% chance of negative sexual side effects.”“Out of every 10 people who take Prozac, 3 to 5 of them develop negative sexual side effects.”Studies show that relative frequencies are easier to think about.
Slide6S
o translate probabilities into relative frequencies!Bonus
benefit: will make you get explicit about the reference class.
Slide7“There’s a 30% chance of rain
today.”“There’s an 80% chance you’ll survive this experimental surgery that’s never been done
before.”
“30% of days like today
rain
.” “I’m 80% sure.”not a probability claim at all, but statement of confidence
Slide8Frequency
t
rees
Suppose there is a 40% chance it will rain today and a 90% chance you’ll get wet if it rains. What is the probability that you get wet today?
Solve
with a frequency tree.
Slide9100
days
40
rains
60
doesn’t rain36get wetTranslate 40% chance of rain into frequencies.Start with a nice round number
.
4
stay dry
90% chance you get wet
if it
rains
.
B
ut
that’s not the only way you might get wet—you might get sprayed by a hose, etc.
even if it doesn’t
rain
.
S
uppose
: 20% chance you get wet
if it doesn’t
rain
.
12
get wet
48
stay dry
Of these 100 days, you get wet on 48 (
36
+
12
) of
them
.
There’s a 48% chance you’ll get
wet
.
Slide10A man between 35 and 44 years old has a 0.6% probability of having prostate cancer. If he has it, there is a 58% chance that the PSA test will catch it. If he doesn’t have it, there is a 23.5% chance that he will test positive nonetheless. X (a
42-year-old man) receives a positive PSA test. What is the probability that he has prostate cancer?
Slide1110,000
men
60
have cancer
9,940 don’t
25test -35test +2336test +7604test -
base rate = .6%
(how many have it regardless of test)
sensitivity of test = 58%
false positives = 23.5%
2371 men test positive (
2336
+
35
)
P
robability
of having cancer
given positive screen
is
35/2371
.
of these, 35 actually
have
it
.
approx. 1.5%!