Sci Ex ch7 – Probability and Uncertainty
Now we discuss how we actually arrive at conclusions, when we have
admitted that we can NEVER show any scientific theory to be true beyond
any reasonable doubt.
“nothing certain but uncertainty”
So how do we “manage” our uncertainty? We use probability and
statistics to tell us whether or not we should be confident of our results,
whether or not we can have reasonable belief in the outcomes of our
experiments.
We use these tools to “talk precisely about impreciseness”

How to gamble and win
Book says prob. was initially developed to help people decide how to bet
in games of chance, where the inability to precisely predict the outcome of
an event is the rule, not the exception.
G&G call this a “state of ignorance”….I think it’s more a reflection of not
having total control of an outcome.

Heads or tails? In a coin toss it’s a 50-50 likelihood that coin comes up
heads or tails… now G&G make a subtle but important point here….
-
One way to frame our uncertainty is to say we have “no reason
to expect one result more than another”, but this makes us
appear to have no ability to predict an outcome…this does us a
disservice because it makes us appear more ignorant than we
really are…
-
Another way to frame this is to ask and answer the question, “
what will happen on average if we toss a coin a large number of
times?” this approach allows us to form a hypothesis and test it.
Note that this approach gives us the power to make very precise
statements about what is likely to happen if we toss the coin, rather than
just giving up and saying “I don’t know” or “I’m not CERTAIN”
This in my mind is important for the non-scientist to “get”…science isn’t
about “certainty”…it’s about reasonableness and determing how we
conclude that an outcome is “reasonable” in light of the data we have at
the time.
Sci Ex ch 7 p.2

Numerical magnitudes - “what are the chances that….” a certain
outcome will happen?
Look at the coin toss again…there’s a .50 or ½ or “1 in 2” or 50% chance
that a coin will come up heads
Another example…what is the chance I’ll draw an ace from a card deck if I
already hold 4 in my hand? (a rare case of 0% probability, and that doesn’t
take into acct the poss of manufacturing and packaging error, like once in
a while there might be 5 aces in a deck…..)
Scale of numerical outcomes ranges from 0 (never)
through 0.5 (even odds)
to 1.0 (absolute certainty)
- Now they discuss independence of outcomes….make good pt about law
of averages…you’d think that as you tossed a coin and kept turning up
heads, the likelihood or probability of getting tails would increase (the “law
of averages” would even things out), but that is NOT the case because the
probability of an outcome of a toin coss is completely independent of other
tosses
- Better example with a card deck….odds of drawing a spade are 13/52 =
.25
if you draw a spade, then draw again, odds of drawing a spade are now
12/51 = .22, slightly less….this is example of a dependent outcome – you
modified the system when you removed that first spade, so your outcome
to some degree is dependent on what you did on your first draw….note
that this is different than flipping a coin

sequence of tosses
- independent probabilities are handled this way:
prob of of two indep events both occurring is the PRODUCT of their
separate probabilities
example , P(2 heads) in 2 tosses = ½ x ½ = ¼ = .25
this makes sense..look at your poss outcomes…
#1 T,#2 H
#1 H,#2 T
#1 T,#2 T
#1 H, #2 H
heads
these are the four outcomes, only one of which has 2
a second rule is used to handle “mutually exclusive” situations like where
we get one or the other but not both (huh?) outcomes…like T,H and
H,T….
Sci Ex ch 7 p.3
- here we add the P(1tail + 1 head) together = ¼ + ¼ = 2/4 = ½
not so sure I like their wording but I think I get their point
another example – toss coin 4 times….what are your poss outcomes?
Table VIII – 5 sets of outcomes , total of 16 possible…
broken up into simple and compound outcomes….the 16 are simple
outcomes, the 5 categories are compound outcomes
P (of any simple outcome) = ½ x ½ x ½ x ½ = 1/16
involving 4 coin tosses
in terms of compound outcomes, there’s only 1/16 for all 4 heads, 1/16 for
all 4 tails...but others a bit more complicated...
1/16 + 1/16 + 1/16 + 1/16 = .25 for any combo of 3 heads, 1 tail or 1 head,
3 tails,
and 6/16 = .375 for any combo of 2H and 2T
interesting pt made by G&G here on 10 coin tosses – no easier to get a
specific sequence of 5 heads and 5 tails than all 10 heads … it just looks
easier (or more probable) because there’s so many combos of 5 heads
and 5 tails …

The Bell Curve
So now we look at the graphic results of “number of heads vs probability
of outcome” for 4 coin tosses
In Fig 29 this is graphed as the compounded number of heads, not the
simple number..0 heads, then 1, 2 , 3, 4…because of the compounded
number of possibilities for 1, 2, and 3 heads, all these probs are > 1/16.
Figs 30 and 31 show results for 10 tosses and 10 tosses
-
whenever a population is present and classified according to a
characteristic that can be measured and divided (like scores on a
midterm) it can be represented by some type of Gaussian distribution
rephrasing what we have learned – what does a prob of 1/1024 mean?
Sci Ex week ch 7 p.4
In this case, it means that we will get , on average after zillions of tosses,
10 heads in a row once every 1,024 times that we toss a coin 10 times in
a row

move on to something more tricky- large no. of balls, white (20%) and
black (80%) in a big barrel
we no longer have equal probability of an outcome, like a head or tail…
P of white ball coming up 4 times in 4 picks is only
.2 x .2 x .2 x .2 = .0016 (16 times out of 10,000)
but P of this with a black ball is
.8 x .8 x .8 x .8 = .4096, or 41 times out of 100 (much better odds)
also note..as your common sense would suggest, that the HIGHEST
probabilities of populations involve a reflection of the real number of balls
in the barrel, that is, 20% white and 80% black are the most likely
outcomes after zillions of draws….

“another” meaning of probability, this time very subjective
here G&G use politics and P of soemone winning an election.
circumstances unique, year unique….
they say we need to look at individual factors, can’t hold 10,000 elections.
so in this context, G&G say that results of tests of sci theories or
hypotheses make the theories either “more” or “less” probable….so I
guess they mean that the results tend to validate or invalidate the theory
Is there a “plate tectonics” link here, like the more modern data, when
coupled with the older data extant at the time of Wegener, would tend to
“increase” the probability of plate tectonics theory being correct??
Sci Ex week ch 7 p.5
turn to their discussion of Application of Prob theory to molecular diffusion
and genetics

molecules in motion – return to Rumford
fairly good description of a common science recipe here…if we theorize or
hypothesize a certain outcome based upon a particular theory, we can
then use probability to determine what the outcome SHOULD be based on
the theory
then you can compare what the results SHOULD be with what they
actually ARE…( I did this a lot in my PhD…)..this may not necess prove
you RIGHT, but it means that you are definitely not wrong…
remember, you can’t get a false answer from true premises….although
you CAN get a true answer from false premises….they ARE different
situations…
- OK , so look at the tortuous path example Fig 35
try and model as strgt line segments, in 3D space, so you have 6 diff poss
of motion along a 3-axis coord system (use your thumb and 2
forefingers…)
up
down
right
left
forward
backward
we can use probability to describe the avg mvmt of a molecule on a path,
based on these 6 outcomes
we determine based on P of outcomes that time t for spreading or
diffusion of salt molecules to occur in water increases with the square of
the size of the vessel holding the solution,
eg, twice the height results in taking 4 times as long to diffuse,
3 times the height takes 9 times as long, etc.
Sci Ex ch 7 p.6
go to the lab, confirm the prediction…so our confidences rises in the
accuracy of our model
we may not be 100% right, but it is also true that we are “probably” not
entirely wrong, either….
 Genetics
look at the “4 o’clock” plant, which is bisexual, so pollen can fertilize flower
of same plant
when 2nd generation pink plants are themselves fertilized, the results are:
red ¼
pink ½
white ¼
Mendel, 1865, published an explanatory hypothesis, disregarded for 35
yrs, then independently re-discovered by 3 diff biols in 1900.
how this relates to the coin toss problem is interesting..Mendel speculates
that there are 2 factors determining flower color – one male parent, one
female parent
but the male pollen cells and the female ova cells each just have one of
the factors
so now what?
look at a pink plant (Fig 37)…when plant forms pollen and ova, each
pollen grain and each ovum cell have either a red or white (like a head or
a tail of a coin)
so then it becomes like the coin flip…how do the pairs work out?
all red (1R, 1R) = ½ x ½ = ¼
pink (meaning 1R, 1W or 1W, 1R) (½ x ½) + (½ x ½) = ½
all white (1W, 1W) = ½ x ½ = ¼
when they move to human genetics, I become lost….
they do reference natural selection at end of chapter, and put it in
framework such as the one I mentioned in class….