Download IGE104-Lecture9

IGE104:
LOGIC AND MATHEMATICS FOR
DAILY LIVING
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Lecture 9: Probability
IGE104 3/2010- Lecture 9
April 1, 2011
THE BIG PICTURE QUESTION

Why is it that we cannot predict the future?
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RANDOMNESS IN DAILY LIFE
Whether
 Car Accidents
 Course grades


How well can we predict these events?
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SIMPLE EXAMPLE: COIN TOSS

What happens if I toss two coins?

List the possible outcomes:

Each row in this table is an outcome

Which outcome would you expect
to occur?
Coin A
Coin B
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SIMPLE EXAMPLE: COIN TOSS

We define an event as one or more possible outcomes
that all have the same property of interest
Coin A

Which outcomes belong to the event
E = one Head ?

What are the chances of having the
event E occur?
Coin B
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THEORETICAL PROBABILITY

When all outcomes are equally likely, we say that the
probability of an event is
Probability (event) = number of outcomes in the event
total number outcomes
We can use set notation to write down the outcomes in
the event.
This is an a priori probability because we calculate it
before we actually see any events.
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THEORETIC PROBABILITY

Ever time we toss a coin there are two possible
outcomes: Heads or Tails

If the coin is “fair” then each outcome is equally likely

So the probability of tossing one head is just 1 out of 2
Probability(heads) = 1 / 2
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WHAT DOES THE NUMBER MEAN?

We saw that the probability of tossing one Head is ½,
which is the same as 0.5…what does this mean?

Well you can think of a probability as a percent. So a
probability of ½ is the same as something being 50%
likely to occur.
What does a probability of zero represent?
 What does a probability of one represent?

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CALCULATING PROBABILITIES WITH EQUALLY LIKELY
OUTCOMES

If we assume that a person is equally likely to be born in
any month of the year, what is the probability that a
random person will have a birthday in April?

Pr(April birthday) = ?
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IGE104 3/2010- Lecture 9
April 1, 2011
CALCULATING PROBABILITIES WITH EQUALLY LIKELY
OUTCOMES

If we assume that a person is equally likely to be born in
any dayof the year, what is the probability that a random
person will have a birthday in April?

Pr(April birthday) = 31 = ~0.08219
365
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CALCULATING PROBABILITIES WITH EQUALLY LIKELY
OUTCOMES

Example: What is the probability that a family with 3
children has 2 girls and one boy?

Pr(2 girls and 1 boy)= ?
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CALCULATING PROBABILITIES WITH EQUALLY LIKELY
OUTCOMES

Example: What is the probability that a family with 3
children has 2 girls and one boy?

Pr(2 girls and 1 boy)= 3 = 0.375
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COUNTING: THE MULTIPLICATION RULE

How to find to total number of possible outcomes.
Previously we had 3 children, where each child could be
either a boy or a girl.
 The total possible number of outcomes was 2 x 2 x 2 = 8
Tree Diagram:
Place Diagram:

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COUNTING: THE MULTIPLICATION RULE

The multiplication rule: the total number of possibilities
is just the product of the number of possibilities for each
“trial”.
Example:
 A restaurant menu offers two choices for an appetizer,
five choices for a main course, and three options for a
dessert. How many different three-course meals are
possible?
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USING THE COUNTING RULE
Suppose you roll one die 10 times. What is the
probability that you get a 1 on the first roll and a 6 on
the next 9 rolls?
610 = 60466176
1/ 610 = 0.000000016538

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COUNTING RULE: BE CAREFUL

Suppose you toss one coin 10 times. What is the
probability that you toss 1 head and 9 tails?
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EMPIRICAL PROBABILITY

Up until now we have practiced finding the theoretical
probability of an event.

But what if we want to find the probability of an event
that we have already observed in the real world?
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EMPIRICAL COIN TOSS
Consider an experiment where we toss two-coins, with
possible events: 0 heads, 1 head, or 2 heads
 Suppose we conduct this experiment 100 times (we toss
two coins 100 times total) and we get the following
results:

0 heads occurs 22 times
 1 head occurs 51 times
 2 heads occurs 27 times


Compared to the theoretical probabilities, do we have
reason to believe that the coins are unfair?
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EXAMPLE: THE 500 YEAR FLOOD

Geological records indicate that a river has crested
above a particular high flood level at least four times in
the past 2,000 years. What is the empirical probability
that the river will crest above this flood level next year?
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PROBABILITY OF AN EVENT NOT OCCURRING

The probability of an event not occurring can be found
using the probability that an event does occur.
Recall from our set notation the relationship between an
event and its negation.
 So Pr(E) + Pr(not E) = 1
 Thus to find Pr(not E),
we can just use
Pr(not E) = 1 – Pr(E)

Not E
E
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IGE104 3/2010- Lecture 9
April 1, 2011
PROBABILITY OF AN EVENT NOT OCCURRING

The probability of an event not occurring can be found
using the probability that an event does occur.
Recall from our set notation the relationship between an
event and its negation.
 So Pr(E) + Pr(not E) = 1
 Thus to find Pr(not E),
we can just use
Pr(not E) = 1 – Pr(E)

Not E
E
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PROBABILITY OF AN EVENT NOT OCCURRING
Probability of not drawing an ace from a full shuffled
deck of cards.
 A deck of cards contains the following numbers
{2♠, 3♠, 4♠, 5♠, 6♠, 7♠, 8♠, 9♠, 10♠, J♠, Q♠, K♠, A♠,
2♣, 3♣, 4♣, 5♣, 6♣, 7♣, 8♣, 9♣, 10♣, J♣, Q♣, K♣, A♣,
2♥, 3♥, 4♥, 5♥, 6♥, 7♥, 8♥, 9♥, 10♥, J♥, Q♥, K♥, A♥,
2♦, 3♦, 4♦, 5♦, 6♦, 7♦, 8♦, 9♦, 10♦, J♦, Q♦, K♦, A♦}
 Pr(not an Ace) = 1 – Pr(Ace) = ?

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SO FAR WHAT WE HAVE FOUND

The probability of an event E
with equally likely outcomes
is just the number of
outcomes that satisfy the
event over the number of all
possible outcomes
Pr(event) =
# outcomes in the event
total outcomes
IGE104 3/2010- Lecture 9

The probability of an
impossible event E is
Pr(E)=0

The probability of an event E
that will definitely occur is
Pr(E)=1

The probability of an event E
not occurring is
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Pr(not E) = 1 – Pr(E)
April 1, 2011
PROBABILITY QUICK CHECK

Can you ever have a probability that is greater than 1?

Can you ever have a probability that is less than 0?
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PROBABILITY DISTRIBUTIONS: TOSSING 3 COINS

Find the probability distribution for the number of heads
that occur when three coins are tossed simultaneously.
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PROBABILITY DISTRIBUTIONS: SUM OF TWO DICE

What is the probability distribution of the possible sums
of two simultaneous dice rolls.
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BIBLIOGRAPHY
(1) Bennett, J. O. , and W. L. Briggs. General Education
Mathematics: New Approaches for a New Millennium.
AMATYC Review, vol. 21, no. 1, Fall 1999, pp. 3-16.

http://math.ucdenver.edu/~wbriggs/qr/AMATYCPaper.html
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