Download stat slides - probabilities

MEASURING PROBABILITIES
ESSENTIAL IDEAS OF PROBABILITY
WHAT’S IT
ABOUT?
The PROBABILITY of an event is a measurement of how possible, or how
likely it is to occur, expressed as a fraction, a decimal, or a percent.
EXAMPLE 1. Let’s consider the simplest case: imagine we are tossing a coin once.
A. For this simple activity, how many possible outcomes can we get?
B. What is the probability of getting a ‘head’ after a toss?
C. What is the probability of getting a ‘tail’ after a toss?
TERMS TO
REMEMBER
Two – ‘head’ and ‘tail’!
1/2 or 0.5 or 50%
1/2 or 0.5 or 50%
An activity that gives us a set of random results or OUTCOMES is called
a PROBABILITY EXPERIMENT (or simply an EXPERIMENT). This set of all possible results is called the SAMPLE SPACE.
A description for a certain set of outcomes is called an EVENT.
MEASURING PROBABILITIES — ESSENTIAL IDEAS OF PROBABILITY
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FORMULA
The PROBABILITY of an event can be computed using the formula:
P(Event ) 
no. of outcomes in the Event
total no. of outcomes in the Sample Space
The notation: P(Event) should be read as “the probability of (Event)”
EXAMPLE 2. A box contains 8 balls labelled 1,2,3, …, 8. Then one ball is picked randomly.
A. List the sample space of the experiment.
SS: 1, 2, 3, 4, 5, 6, 7, 8
B. List the outcomes included in the following events.
Event A = The ball picked is an odd number. = 1, 3, 5, 7
Event B = The ball picked is less than 4.
= 1, 2, 3
Event C = The ball picked is greater than 9
= (none)
Event D = The ball picked is less than 10
= 1, 2, 3, 4, 5, 6, 7, 8, 9
C. Compute the following probabilities.
P(The ball picked is an odd number) = 4/8 or 0.5 or 50%
P(The ball picked is greater than 9) = 0/8 or 0 or 0%
P(The ball picked is less than 10) = 8/8 or 1 or 100%
MEASURING PROBABILITIES — ESSENTIAL IDEAS OF PROBABILITY
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EXAMPLE 3. One card is randomly drawn from a standard deck of playing cards.
A. List the sample space of the experiment.
SS: A♣ A♠ A♥ A♦ 2♣ 2♠ 2♥ 2♦ . . . K♣ K♠ K♥
K♦
Count? — 52
B. Compute the following probabilities.
P(Drawing a Queen) = 4/52
P(Drawing a Four of Clubs or a Nine) = 5/52
P(Drawing a Jack or a King) = 8/52
P(Drawing a Seven or a Heart) = 16/52
EXAMPLE 4. A box contains eight balls, labelled 1, 2, 3, …, 8. Two balls are picked at once.
A. List the sample space of the experiment.
SS: 1,2 1,3 1,4
3,5 3,6 3,7
Count? — 28
NOTE!
1,5
3,8
1,6
4,5
1,7
4,6
1,8
4,7
2,3
4,8
2,4
5,6
2,5
5,7
2,6
5,8
2,7
6,7
2,8
6,8
3,4
7,8
‘AT ONCE’: picking a group of items, all at the same time.
Here, the items picked have no ordering. Hence, the pairs 1,2 and 2,1 are the same!
MEASURING PROBABILITIES — ESSENTIAL IDEAS OF PROBABILITY
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EXAMPLE 6. Two balls are picked with replacement from a box containing six balls, labelled
1, 2, 3, 4, 5, 6.
A. List the sample space of the experiment.
SS: 1,1
3,1
5,1
1,2
3,2
5,2
1,3
3,3
5,3
1,4
3,4
5,4
1,5
3,5
5,5
1,6
3,6
5,6
2,1
4,1
6,1
2,2
4,2
6,2
2,3
4,3
6,3
2,4
4,4
6,4
2,5
4,5
6,5
2,6
4,6
6,6
Count? — 36
NOTE!
‘WITH REPLACEMENT’: one by one, but returning an item before picking the next
Here, the items picked have an ordering! Hence, the pairs 2,4 and 4,2 are not the same!
B. Compute the following probabilities.
P(Drawing a sum of at least 8) = 15/36
P(Drawing a difference of at most 2) = 24/36
P(Drawing a difference of at least 6) = 0/36 or 0
REMEMBER!
If P(Event) = 0, there is no possibility for the event to occur.
If P(Event) = 1, the event is absolutely sure to occur.
The percent (%) value of the probability gives us the estimated number
of times the event will occur after 100 repetitions of the experiment.
MEASURING PROBABILITIES — ESSENTIAL IDEAS OF PROBABILITY
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EXAMPLE 4. A bowl contains 100 identical-looking chocolate candies of four kinds: either milk or
dark chocolate with either a nut or a raisin filling. 40 of them are milk chocolate, 46
have raisin filling and 26 are dark with raisin filling. One candy is picked randomly.
A. List the sample space of the experiment.
(We cannot list the sample space. Therefore, we will use a drawing.)
40
milk
100
candies
60
dark
20
w/ nuts
20
w/ raisins
34
w/ nuts
46
26
w/ raisins
B. Compute the following probabilities.
P(Getting a dark chocolate candy) = 60/100
P(Getting a candy with nut filling) = 54/100
P(Getting a dark chocolate or a candy with nuts) = 80/100
P(Not getting a dark chocolate or a candy with raisins) = 54/100
MEASURING PROBABILITIES — ESSENTIAL IDEAS OF PROBABILITY
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PROBABILITIES FROM ACTUAL EXPERIMENTS
EXAMPLE 6. At the end of SY 2015-2016, a Comprehensive Assessment Exam in Algebra was conducted among all 500
freshman students of Recto High with the ff results:
A. A freshman student is randomly chosen in Recto High. Compute
the following probabilities.
P(The student got an CAE rating within 59.5–79.5) = 229/500
P(The student got an CAE rating above 79.5) = 102/500
P(The student got an CAE rating below 49.5) = 72/500
B. Would these probability values remain the same, say, next year?
WHAT’S IT
ABOUT?
EXAM RATING FREQ.
19.5 – 29.5
5
29.5 – 39.5
26
39.5 – 49.5
41
49.5 – 59.5
97
59.5 – 69.5
124
69.5 – 79.5
105
79.5 – 89.5
67
89.5 – 99.5
35
Probability values calculated from actual experiments are called
EMPIRICAL PROBABILITIES, while those that come from a complete
description of the sample space and the relevant events are called
THEORETICAL PROBABILITIES (which is the ‘true’ probability value).
MEASURING PROBABILITIES — PROBABILITIES FROM ACTUAL EXPERIMENTS
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EMPIRICAL PROBABILITY REACHES THE ‘TRUE’ VALUE, IN TIME!
Simulation of the experiment of tossing a die (120000 times) in Microsoft Excel
120000 RANDOM NOS. FROM 1-6
4
5
5
1
4
6
6
4
6
6
3
4
3
5
4
6
4
1
1
6
NOTE!
3
1
3
6
1
1
2
4
4
3
3
1
4
3
4
2
4
4
2
1
5
6
3
2
6
2
4
3
5
6
3
4
4
3
3
2
1
5
1
1
1
5
4
1
1
1
2
1
5
4
3
2
2
1
2
1
5
1
3
5
4
1
3
4
5
3
1
4
6
2
3
4
6
6
1
2
5
1
2
4
3
5
6
2
3
2
3
5
4
4
5
2
3
5
2
6
6
3
1
1
6
4
1
3
1
3
3
3
4
6
5
4
2
3
1
2
1
3
5
5
5
5
1
4
2
5
2
2
4
3
2
1
6
6
5
3
4
4
1
1
4
2
4
4
2
1
4
6
6
4
2
3
5
1
3
2
6
5
4
4
3
6
5
3
5
5
3
4
6
3
4
4
1
3
5
4
6
4
4
2
NO.
FREQ.
R.FREQ
1
2
3
4
5
6
19963
19855
20019
20042
20034
20087
0.166358
0.165458
0.166825
0.167017
0.16695
0.167392
TOTAL
120000
Repeating the experiment a large number of times makes the calculated probability
values approach the respective theoretical or ‘true’ probability values.
MEASURING PROBABILITIES — EMPIRICAL PROBABILITIES REACH THE ‘TRUE’ VALUE, IN TIME!
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