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Notes From David Palay:
Chapter 5.1
Introduction to Probability
What are the chances that…
Probability
• From the book,
– “The probability of an outcome is defined as the longterm proportion of times the outcome occurs.”
• From Wikipedia,
– “Probability is a way of expressing knowledge or
belief that an event will occur or has occurred.”
• Mr. David Palay,
– “Probability is the chance something will or will not
happen”
Terms
• Experiment
– An activity where the outcome is uncertain
• NOT NECESSARILY UNKNOWN, JUST UNCERTAIN
• Outcome
– Result of a single trial of an experiment
• Sample Space
– Collection of all possible outcomes of an experiment
• Event
– Collection of outcomes from the sample space of an
experiment
Rules of Probability
• We write the probability of an event E as 𝑃 𝐸
• 0≤𝑃 𝐸 ≤1
– Which means that the probability of any event
is between 0 and 1.
• 0 means it will NEVER EVER EVER EVER HAPPEN.
• 1 means it will ALWAYS happen
Are these valid probabilities?
𝑃(𝐴) = .403
𝑃(𝐵) = .32384
𝑃(𝐶) = 2.32 ∗ 10−3
𝑃 𝐷 = −3.21
𝑃 𝐸 = 4.324
Rules of Probability (continued)
• For any given experiment, the probability of
the sum of the outcome probabilities in the
sample space must equal 1.
– SOMETHING has to happen, or we have an
incomplete sample space.
Experiment & Theory
• Experimental Probability:
– Also called the “relative frequency method”
– Probability we get from the results of running
tests.
• Theoretical Probability:
– Also called the “classical method”
– The probability calculated based on the rules of
mathematical probability. (Which we will touch
on later)
Dice nomenclature
𝑥d𝑦 – read “x dee y”, represents throwing x fair
dice, each with y sides.
e.g.,
• 3d6 (“three dee six”) represents rolling 3 six
sided dice.
• 1d20: 1 twenty sided die
Some Examples
Experiment
Roll 1d6
Sample Example Events
Space
•
•
•
Rolling a six: {6}
Rolling an even number: {2, 4,
6}
Rolling under a 3: {1,2}
Flip two coins
•
•
Randomly
pick a billiard
ball
• Picking a solid: {1, 2, 3, 4,
5, 6, 7, 8}
• Picking a yellow ball {1, 9}
• Picking the 8-ball {8}
Getting 2 heads {HH}
Getting at least 1 head
{HH, HT, TH}
Basic Probability
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐸 𝑐𝑎𝑛 ℎ𝑎𝑝𝑝𝑒𝑛
𝑃 𝐸 =
𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Ok, that sounds easy..
Find:
P(rolling a 3 on 1d6):
P(rolling odds on 1d6):
Which is greater? Why?
More Practice
• Standard deck of cards: 4 suits {Spades,
Diamonds, Hearts, Clubs} and 2-10, Ace, Jack,
Queen, King. The Jack, Queen, and King are
considered “Face cards”
P(drawing a 3 from a shuffled deck):
P(drawing a face-card of hearts):
Slightly harder now…
• What single sum has the highest probability of
coming up when we roll 2d6?
• We need to figure out how many possibilities
there are.
– Ah HA! Counting! We have 2 “spots”, each with 6
possibilities. So….
–𝑃 𝐸 =
?
36
2d6 continued
1
1
2
3
4
5
6
2
3
4
5
6
So, we can see…
𝑃 2 =
𝑃 3 =
𝑃 4 =
𝑃 5 =
𝑃(6) =
𝑃 7 =
1
36
2
36
3
36
4
36
5
36
6
36
=
=
=
=
1
18
1
12
1
9
1
6
5
𝑃 8 =
36
4
1
𝑃 9 = =
36
9
3
1
𝑃 10 = =
36
12
2
1
𝑃(11) = =
36
18
1
𝑃 12 =
36
Law of Large Numbers
Given a sufficiently large number (infinite) of
trials, the Experimental Probability will approach
the Theoretical Probability
The Great Glass Rod Problem
• If we take a glass rod, and break it at two
random points, what is the probability that we
will be able to form a triangle with the pieces.
Subjective Probability
• Intuition. Guessing. Personal Judgement.