Probability
Probability
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Probability
The study of probability dates back to the mid 17th century through
correspondence between two mathematicians, Pierre de Fermat and
Blaise Pascal.
Here is a quote from Calculus, Volume II by Tom M. Apostol (2nd
edition, John Wiley & Sons, 1969): “A gambler’s dispute in 1654 led
to the creation of a mathematical theory of probability by two famous
French mathematicians, Blaise Pascal and Pierre de Fermat.
Antoine Gombaud, Chevalier de Méré, a French nobleman with an
interest in gaming and gambling questions, called Pascal’s attention
to an apparent contradiction concerning a popular dice game.The
game consisted in throwing a pair of dice 24 times; the problem was
to decide whether or not to bet even money on the occurrence of at
least one double six during the 24 throws.
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A seemingly well-established gambling rule led de Méré to believe that
betting on a double six in 24 throws would be profitable, but his own
calculations indicated just the opposite.
This problem and others posed by de Méré led to an exchange of
letters between Pascal and Fermat in which the fundamental
principles of probability theory were formulated for the first time.
Although a few special problems on games of chance had been solved
by some Italian mathematicians in the 15th and 16th centuries, no
general theory was developed before this famous correspondence.”
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Simulation of de Méré’s Problem
Roll a pair of dice 24 times. If you roll double six at least once, then
you win. If you never get double six on any of the 24 rolls, you lose.
Once you are done, submit
A Win
B Lose
The clicker software will show what percentage of the class wins.
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It turns out that the probability of losing is
(35/36)24
which is approximately .51, or about 51%.
Then the probability of winning is approximately .49, or slightly less
than half. Thus, de Méré was right that it is a little less than even
money to bet on getting a double six in 24 rolls. Later on we will see
how to come up with this calculation.
Doing the computation above requires a scientific calculator. There
are free websites that do the same thing. One is
http://web2.0calc.com
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If you have a smart phone you may already have a scientific calculator.
For example, the iPhone comes with a calculator app. Rotating the
phone toggles between a regular and scientific calculator.
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Simulating de Méré’s Problem with Excel
We will use the Microsoft Excel spreadsheet deMereProblem.xlsx to
simulate our experiment. The advantage is that we can conduct
many trials in a short amount of time; many more than we could do
by actually rolling dice.
This spreadsheet, and all others we will use, will be on the course
websites.
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Theoretical and Experimental Probability
This week we will explore some of the ideas that are used in this
calculation. But first, we will focus on getting a better understanding
of what is the meaning of probability.
If you conduct an experiment, such as the dice rolling we did earlier,
you can compute an experimental probability, as we did. Our
percentage of wins for the entire class is an experimental probability
for winning that game.
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If we say the probability of getting heads when you flip a coin is 50%,
this is a theoretical probability. It is telling us what we expect to get
if we flip a bunch of coins.
The reality is a little more complicated, as we’ll explore. Roughly, to
say that the probability (or chance) of flipping a coin and getting
heads is 50%, or .5, then on average, if you flip a coin many times,
you will expect 50% of the flips being heads. However, what happens
in a given set of flips can be nearly anything.
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The theoretical probability allows us to estimate what will happen
when we conduct an experiment. However, we can use an
experimental probability to estimate theoretical probability. Therefore,
each can be used to estimate the other.
For example, casinos can estimate how much money they will make
from a game by knowing the theoretical probability of winning that
game. What actually happens on a given day is an experimental
probability. Because they have so many people playing, their
estimates are usually pretty good.
There is always a chance that somebody wins big on a given day.
But, if they estimate income for longer periods of time, they’ll be
even more accurate.
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A Coin Flip Experiment
Flip a coin once and record the number of heads (either 0 or 1) with
your clicker.
We will show the class data.
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Another Coin Flip Experiment
Flip a coin 20 times and enter the number of heads you got.
We will tabulate the data for the entire class. What does the data
indicate to you about the probability of getting a heads on a flip of a
coin?
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Simulating the Coin Flip Experiment with Excel
We will use the spreadsheet Coin Flip.xlsx to simulate flipping a coin.
We will simulate flipping different numbers of coins with the
spreadsheet. One thing to think about is how do you think the
number of coins we flip will affect the results. How did flipping one
coin versus 20 coins affect our results in the previous experiment?
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Observations from the Simulation
How did the variability of the percentage of heads change as we
increased the number of flips?
The percentage of heads didn’t change as much from trial to trial
when we flipped more coins. We are therefore getting a better
estimate of the theoretical probability the more coins we flip.
Do you think the spreadsheet confirmed that the probability is 50% to
get a heads (assuming the spreadsheet does correctly simulate flipping
a coin)?
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A Final Coin Flip Experiment
If you flip two coins, do you think it is equally likely to get two heads
as it is to get a heads and a tails? When you flip two coins, you can
get two heads, two tails, or one of each.
Flip a pair of coins 20 times and record how many times you got 2
heads and how many times you got one of each. You don’t need to
keep track of how many times you got 2 tails.
Enter the number of times you got 2 heads with your clicker.
Now enter the number of times you got one of each.
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Are you surprised by the class’s results? It turns out that the
probability of getting one of each is twice as much as getting two
heads. The probability of getting one of each is 50% while the
probability of getting two heads is 25%.
Because we can view this experiment as having three outcomes,
2 heads,
2 tails,
one of each
many people would expect each to have a 1/3 chance of happening.
We will now simulate this activity with a spreadsheet, Two Flips.xlsx.
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Calculating Probabilities
We’ve looked at several probability experiments and have computed
some experimental probabilities. How does one go about calculating
theoretical probabilities? For example, what calculation did de Méré
do to see that the rolling a pair of dice 24 times was slightly less than
50/50 to get a double six?
We’ll start to look at some ideas for computing probability. The first
idea we will look at is the basis of most probability computations, and
virtually all we will consider.
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Equally Likely Events
A very common situation in probability is to have all possible
outcomes be equally likely. Examples are flipping a (single) coin and
rolling a (single) die. The two outcomes of the flip, heads and tails,
each have a 50% chance of occurring. Each of the 6 outcomes of
rolling a die have a 1/6 chance of happening.
When we have equally likely events, computing probability is simpler.
To compute the probability that something happens, you can then use
the formula
probability =
Probability
number of ways the outcome can occur
.
total number of outcomes
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An Example using a Spinner
Respond to each question by entering a decimal with your clicker.
1. If you spin this spinner, what is the probability you land on blue?
1/5 = .2
2. What is the probability you land on yellow?
2/5 = .4
3. What is the probability you don’t land on red?
4/5 = .8
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If we interpret an outcome as landing in one of the five wedges, then
there are 5 equally likely outcomes. With this interpretation, to
answer the third question, we have to recognize that 4 of the
outcomes represent not landing on red. That is, not landing on red is
the same as landing on either blue, yellow, or green.
A perfectly good interpretation would be to list outcomes as the
different colors. If we do this, then the outcomes are not equally
likely. It is usually much simpler to interpret outcomes in a way to
make them equally likely (if possible).
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Listing Outcomes
If you can model a situation with equally likely outcomes, then writing
out all outcomes is the most basic way to compute probabilities, but
is often effective.
For example, if you flip a single coin, you have two equally likely
outcomes:
heads, tails
If you roll a single die, you have six equally likely outcomes:
1, 2, 3, 4, 5, 6
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Many situations in probability can be viewed as multi-stage situations.
For example, when flipping two coins, you can think about it as
flipping one, then flipping the other (even if you flip simultaneously).
Similarly rolling two dice can be viewed as a two-stage situation. The
problem we started to discuss on Monday, rolling a pair of dice 24
times, can be viewed as a 24-stage situation (or a 48-stage!).
For example, if we flip two coins, we can list the outcomes as
HH,
HT ,
TH,
TT
Where H = heads and T = tails and by thinking of HT as flipping
heads, then tails, while TH represents flipping tails, then heads.
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We could also model this situation by writing the outcomes as
two heads,
two tails,
one of each
While this is fine, the outcomes are not equally likely, as our
simulations have shown. It is for this reason that the first way of
modeling flipping two coins is more convenient.
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The situation of rolling two dice is similar, but there are more
outcomes. If we think of this as a two-stage situation, rolling the first
die, then the second, we can list the outcomes in several ways. One
way is with the following table.
PP
PP
die 1
die 2
PP
PP
P
1
2
3
4
5
6
Probability
1
2
3
4
5
6
(1, 1)
(2, 1)
(3, 1)
(4, 1)
(5, 1)
(6, 1)
(1, 2)
(2, 2)
(3, 2)
(4, 2)
(5, 2)
(6, 2)
(1, 3)
(2, 3)
(3, 3)
(4, 3)
(5, 3)
(6, 3)
(1, 4)
(2, 4)
(3, 4)
(4, 4)
(5, 4)
(6, 4)
(1, 5)
(2, 5)
(3, 5)
(4, 5)
(5, 5)
(6, 5)
(1, 6)
(2, 6)
(3, 6)
(4, 6)
(5, 6)
(6, 6)
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Here is another way to list the 36 outcomes of rolling two dice, and
keeping track that we are considering one die as the first die and the
other as the second.
(1,1)
(1,2)
(2,1)
(2,2)
(3,1)
(3,2)
(4,1)
(4,2)
(5,1)
(5,2)
(6,1)
(6,2)
Probability
(1,3)
(1,4)
(2,3)
(2,4)
(3,3)
(3,4)
(4,3)
(4,4)
(5,3)
(5,4)
(6,3)
(6,4)
(1,5)
(1,6)
(2,5)
(2,6)
(3,5)
(3,6)
(4,5)
(4,6)
(5,5)
(5,6)
(6,5)
(6,6)
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Next Time
We will look at ways of calculating some theoretical probabilities.
This will allow us then to estimate experimental probabilities.
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