Uncertainty 17: The binomial
distribution
Previous clip:
Two outcomes, either
1) 2,3,10,11 or 12 on two dice = “success”
2) 4,5,6,7,8,9 on two dice
= “failure”
Previous clip:
Two outcomes, either
1) 2,3,10,11 or 12 on two dice = “success”
2) 4,5,6,7,8,9 on two dice
= “failure”
Success:
and
and
and
and
and
and
Previous clip:
Two outcomes, either
1) 2,3,10,11 or 12 on two dice = “success”
2) 4,5,6,7,8,9 on two dice
= “failure”
Models: p=Pr(“success” on any given event)
for independent events
1) p1=Pr(success | M1)=1/4
2) p2=Pr(success | M2)=5/11
Previous clip:
Two outcomes, either
1) 2,3,10,11 or 12 on two dice = “success”
2) 4,5,6,7,8,9 on two dice
= “failure”
Models: p=Pr(“success” on any given event)
for independent events
1) p1=Pr(success | M1)=1/4
2) p2=Pr(success | M2)=5/11
Observed after 100 trials: 24 successes.
Previous clip:
Two outcomes, either
1) 2,3,10,11 or 12 on two dice = “success”
2) 4,5,6,7,8,9 on two dice
= “failure”
Models: p=Pr(“success” on any given event)
for independent events
1) p1=Pr(success | M1)=1/4
2) p2=Pr(success | M2)=5/11
Observed after 100 trials: 24 successes.
Observed success rate: 0.24
:
Likelihood ratio:
24
76
Pr(D|M1)
p1 (1-p1)
------------ = ------------- = 19000
24
76
Pr(D|M2)
p2 (1-p2)
:
Likelihood ratio:
24
76
Pr(D|M1)
p1 (1-p1)
------------ = ------------- = 19000
24
76
Pr(D|M2)
p2 (1-p2)
Bayes formula:
Pr(M1)Pr(D|M1)
Pr(M1 | D ) = -------------------------------------------- = 99.953%
Pr(M1)Pr(D|M1)+Pr(M2)Pr(D|M2)
(for Pr(M1=Pr(M2)=1/2)
Pr(M2 | D)=1-Pr(M1 | D)=0.047%
'k' successes out of 'n' trials.
'k' successes out of 'n' trials.
Pr(k successes out of n trials in a given ordering)=
pk * (1-p)n-k
where p=Pr(success on single trial)
'k' successes out of 'n' trials.
Pr(k successes out of n trials in a given ordering)=
pk * (1-p)n-k
where p=Pr(success on single trial)
k out of n in a given ordering mutually exclusive
from k out of n in another ordering.
'k' successes out of 'n' trials.
Pr(k successes out of n trials in a given ordering)=
pk * (1-p)n-k
where p=Pr(success on single trial)
k out of n in a given ordering mutually exclusive
from k out of n in another ordering.
Pr(k out of n in a given ordering or k out of n in
another given ordering)=
Pr(k out of n in a given ordering)+
Pr(k out of n in a another given ordering) =
2*pk * (1-p)n-k
'k' successes out of 'n' trials.
Pr(k successes out of n trials in a given ordering)=
pk * (1-p)n-k
where p=Pr(success on single trial)
Pr(k out of n successes in any ordering)=
pk * (1-p)n-k * number of orderings of k successes
and n-k failures.
Number of ways to order k successes
from n trials
Number of ways to order n items:
for n=3, there are 3*2*1=6 ways:
Number of ways to order n items:
n items to put in first place times
n-1 items to put in the second place times
...
...
...
2 items to put in the second last place times
1 item to put in the last place
Number of ways to order n items:
n items to put in first place times
n-1 items to put in the second place times
...
...
...
2 items to put in the second last place times
1 item to put in the last place
= n*(n-1)*...*2*1
Number of ways to order n items:
n items to put in first place times
n-1 items to put in the second place times
...
...
...
2 items to put in the second last place times
1 item to put in the last place
= n*(n-1)*...*2*1
=n! (n factorial)
Number of ways to order n items
with two identical items:
for n=3, there are 3*2*1/2=3 ways:
Number of ways to order n items with
containing k identical items
=
number of ways to order n different items
-------------------------------------------------------
number of ways to order the k identical items if they were different
= n! / k!
Number of ways to order n items with
containing k identical items and n-k other
identical items
n!
= ------------k! (n-k)!
Number of ways to order n items with
containing k identical items and n-k other
identical items
n!
(def)
= ------------- =
k! (n-k)!

n
k
Binomial distribution:
Pr(k successes from n trials)=

n p k 1− pn−k
k
Binomial distribution
(probability of getting k
successes in n trials),
for n=100 trials
Peak value
Binomial distribution
(probability of getting k
successes in n trials),
for n=100 trials
Observed
outcome
Peak value
Binomial distribution
(probability of getting k
successes in n trials),
for n=100 trials
Peak value
Binomial distribution
(probability of getting k
successes in n trials),
for n=100 trials
Most probable outcomes
Outcome distribution
for model 1, p=1/4
Outcome distribution
for model 1, p=1/4
Outcome distribution
for model 2, p=5/11
Evidence for
model 1
Outcome distribution
for model 1, p=1/4
Outcome distribution
for model 2, p=5/11
Evidence for
model 1
Outcome distribution
for model 1, p=1/4
Evidence for
model 2
Outcome distribution
for model 2, p=5/11
Outcome distribution
for model 1, p=1/4
Outcome distribution
for model 2, p=5/11
Marked in red:
Probability of
getting evidence
for model 2, when
the data has been
produced by
model 1.
Outcome distribution
for model 1, p=1/4
Outcome distribution
for model 2, p=5/11
Marked in red:
Probability of
getting evidence
for model 2, when
the data has been
produced by
model 1 = 1.64%.
Outcome distribution
for model 1, p=1/4
Probability of
getting
evidence
for model 1
given
that it has
produced
the data =
98.36%.
Outcome distribution
for model 2, p=5/11
Marked in red:
Probability of
getting evidence
for model 2, when
the data has been
produced by
model 1 = 1.64%.
Why n=100 trial?
Wouldn't something else
function just as well?
Model 1
probabilities
Most probable
outcomes for model 1.
Most probable
outcomes for model 1.
Most probable
outcomes for model 2.
Outcomes k successes out
of n trials means an
observed rate of k/n
Most probable outcomes
between 0.15 and 0.35.
Most probable outcomes
for model 1 between 0
and 0.5.
Most probable outcomes
for model 1 between 0
and 0.5.
Most probable
outcomes for
model 2 between
0.2 and 0.8.
Marked in red:
Pr(evidence for model 2 | model 1)=
Pr(k/n=0.4 | p=1/4) +
Pr(k/n=0.5 | p=1/4) +
Pr(k/n=0.6 | p=1/4) +
Pr(k/n=0.7 | p=1/4) +
Pr(k/n=0.8 | p=1/4) +
Pr(k/n=0.9 | p=1/4) +
Pr(k/n=1.0 | p=1/4) =
22.4%
Pr(evidence for model 2 | model 1)
= Pr(k/n=347/1000 | p=1/4) +
Pr(k/n=348/1000 | p=1/4) + ... +
Pr(k/n=1 | p=1/4) ≈ 3.2 *10-12