# Wollongong 2008.ppt.pot

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```					                      Probability!                                             Review by T Tao Secondary 2B

Probability is a notoriously subtle
subject, full of pitfalls from sloppy
Michael Evans
reasoning or loose wording, but the
michael@amsi.org.au
authors have been particularly careful
here to make the material here
International Centre of Excellence for Education in Mathematics   correct, unambiguous, and clear, in
Australian Mathematical Sciences Institute
111 Barry Street, Carlton, VIC
particular relying on visual aids such
as arrays and trees to facilitate
intuition.

Review by T Tao Secondary 2B
Bertrand's box paradox is a classic
(At the tertiary level, one would want                            paradox of elementary probability
to challenge the students with the                                theory. It was ﬁrst posed by Joseph
(many) probability paradoxes and                                  Bertrand in his Calculations of
other subtleties in order to test their                           Probabilties published in 1889.

grasp of the subject, but one should
There are three boxes: a box
of course avoid this for more junior
containing two gold coins, a box with
students who are only just
two silver coins, and a box with one
encountering the subject for the first
of each.

time.)

After choosing a box at random and                                   Probability has many applications
withdrawing one coin at random that                                  • Medicine
• Cryptanalysis
happens to be a gold coin what is the                                • Speech recognition
probability that the remaining coin is                               • Biology
gold?

1
“Using statistics and probability to discover
What’s happening in Mathematics
disease-causing genes in human pedigrees”

Melanie Bahlo works in statistical genetics
University of Wollongong                                 and bioinformatics.
August 13
This involves the statistical analysis

of DNA data to identify genetic loci involved in disease.

She collaborates with medical scientists, working

on diseases as diverse as epilepsy, multiple sclerosis

and deafness.

She enjoys the immediate application of mathematics

to biological problems, and seeing her work translate
to medical benefits for the public.

Pierre Fermat (1601 - 1665)

History of Probability
Probability had a late start compared to other
areas such as Mechanics or Geometry.
The Greeks developed no theory
The first formulation of the principle of symmetry
of dice game was made in a work of Facio
Cardano (1444 - 1524)
The work was found amongst his papers long
after his death and published in 1663.

Blaise Pascal (1623 - 1662)
Chevalier de Mere. Around 1650,
he suffered severe financial losses for assessing
incorrectly his chances of winning
in certain games of dice. Contrary to the ordinary gambler,
he pursued the cause of his error with the help of Blaise Pascal.
Among other things, the Chevalier systematically
tried his luck with the following two games.

2
First game: Roll a single die 4 times and bet on          Second game: Roll two dice 24 times and bet on
getting a six.                                            getting a double six.
He assessed his chances of winning as follows.            He assessed his chances of winning as follows.
The chance of getting a 6 in a single throw is 1 out of   The chance of getting a double six in one roll is 1 out
6.                                                        of 36.
Therefore, the chances of getting a double six in 24
He assessed his chances of winning as follows.            rolls is 24 times 1 out of 36; i.e. 2 out of 3
The chance of getting a 6 in a single throw is 1 out of
6.
Therefore, the chance of getting a 6 in 4 rolls is 4
times 1 out of 6; i.e. 2 out of 3.

Marquis de Laplace (1749-1827)
wrote the book, Analytical Theory in                                  A.  N. Kolmogorov (1903 - 1987) wrote:
Probability, where he presented 10
principles of probability calculations                                ‘ Foundations of the theory of Probability’
as a general introduction and then                                        in 1933
he went on to apply these to natural
philosophy and moral sciences.                                        An excellent reference for probability is
‘An introduction to Probability Theory and
its Applications’ William Feller,J Wiley
and Sons, 1964

Teaching Probability in years 7 to 10.                    Teaching Probability in years 7 to 10.

Probability by the principle of                           Probability by the principle of
symmetry.                                                 symmetry.
Offer questions which are reducible to
considering a sample space with                        In this case if A is any event in S
equally likely outcomes                                                P(A) = r
n
S is a finite sample space with                           Where r is the number of outcomes in A
S = { a1, a2, …, an} where the ai are
the outcomes of the experiment and
P(ai) = p for all i.

3
Teaching Probability in years 7 to 10.

Teaching Probability in years 7 to 10.

Probability by the principle of
symmetry.

Up to the end of year 9 the
emphasis should be on listing sample
spaces using techniques such as tree
diagrams, Venn diagrams and arrays.

Teaching Probability in years 7 to 10.              Teaching Probability in years 7 to 10.

Teaching Probability in years 7 to 10.              Teaching Probability in years 7 to 10.

The idea of mutually exclusive occurs naturally

and the probability formulae follow by using
Venn diagrams
Empirical probability (relative frequency) is
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)                 considered but quarantined at this stage.
P(Ac) = 1 – P(A)

4
Teaching Probability in years 7 to 10.   Teaching Probability in years 7 to 10.

Creating a new sample space              Conditional probability

Sampling with and without replacement    Sampling with and without replacement

5
Sampling with and without replacement

6

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