Allen Carandang
MATH 1090
Introduction to Probability
What does it REALLY mean to “draw a sample”?
In this class, we have talked about getting a “sample” for a statistical study. A sample is just a set of
individuals (subjects) drawn from the population. We then study that sample and try to draw conclusions
about the bigger population.
But when we draw a random sample, how can we really know that we are getting a sample that truly
represents the population? What if we want to know what “students” believe about something. Suppose
we select a single student and talk to them. The student is white, male, comes from a wealthy background,
and has always found school to be easy. Is that student going to have views and opinions that represent all
students? No! For one thing, if you only have one student , you will only get a single opinion on any give
topic. Additionally, the cultural influences on that student and their personal history and experiences will
inform their opinions.
Clearly one student is not enough. So, how many is enough? 2? 5? 1,000? 1,000,000? If we have to talk to
a million people every time we want to learn something, it will cost a fortune! Luckily, statistics to the
rescue!
For the next period of time in our class, we are going to tell the story of how statistics, starting from the
concept of the normal curve, is able to figure out how many individuals we need to make sure that the
sample is big enough that it is LIKELY to be representative.
Wait a minute, what do we mean by LIKELY? For example, even if we took a sample of 1,000 students,
isn’t it possible that we could accidentally get 1,000 white, male, wealthy students for whom school is
easy? The answer is YES, it is POSSIBLE. But it is not LIKELY.
In statistics, the measure we use for how LIKELY or UNLIKELY something is to happen is called
“Probability”. In this activity, we’re going to explore basic probability concepts to begin to formalize our
current understanding of what is LIKELY and UNLIKELY.
Allen Carandang
MATH 1090
Let’s Flip a Coin
From our book (Chapter 17):
We call a phenomenon random if individual outcomes are uncertain but there is, nonetheless, a
regular distribution of outcomes in a large number of repetitions.
The probability of any outcome of a random phenomenon is a number between 0 and 1 that
describes the proportion of times the outcome would occur in a very long series of repetitions.
Let’s practice these ideas.
1. If you flip a coin one time, what are the possible outcomes?
Heads or Tails {H, T}
2. If you flip a coin twice, what are the possible outcomes?
{HH, HT, TH, TT}
HH means two heads.
HT means getting a head on the first flip and a tail on the second flip.
TH means getting a tail on the first flip and a head on the second flip.
TT means two tails.
When we talk about probabilities, we are really looking for the proportion of times that an outcome
happens.
The possible outcomes for flipping a coin are Heads (H) and Tails (T). Let’s rephrase the questions above:
1. If you flip a coin one time, there are two possible proportions for the number of Heads in your
flip. What are they?
If you flipped a tail, then the number of heads is 0. The probability of getting 0 heads is P(0) = ½
= .50
If you flipped a head, then the number of heads is 1. The probability of getting 1 head is P(1) = ½
= .50
2. If you flip a coin two times, what are the possible proportions of heads for the various outcomes?
If the outcome is TT, then the number of heads is 0. The probability of getting 0 heads is P(0) = ¼
= .25
If the outcome is HT or TH, then the number of heads is 1. The probability of getting 1 head is
P(1) = 2/4 = .50
If the outcome is HH, then the number of heads is 2. The probability of getting 2 heads is P(2) =
¼ = .25
3. In the long run, if you flipped a coin a LOT of times, what would you expect the proportion of
heads to be? Why?
I would expect the proportion of heads to be ½ = .50 = 50% sine there are two possible outcomes
when flipping a coin once.
Allen Carandang
MATH 1090
Let’s simulate doing these coin flips. Open the Statistical Applet located at
https://www.statcrunch.com/applets/type3&coins. Leave it open in a separate browser window as you
work through this assignment together.
1. Click on word “Outcomes” in the simulator. Then click on “1 flip” a few times. How many heads
did you get?
I got one head
2. Click on “reset” and then click on “Convergence”. Complete the table below by clicking on “1
flip” each time to find the cumulative proportion:
Flip
#
Proportion
of Heads
1
1
2
1
3
.67
4
.5
5
.4
6
.33
7
.43
8
.38
9
.44
10
.4
11
.36
12
.42
13
.38
14
.36
15
.33
16
.38
17
.35
18
.33
19
.37
Allen Carandang
MATH 1090
20
.35
3. What do you see happening?
As the amount of flips increase, the proportion stays within the range of .33 to .43.
4. Reset the simulation by clicking the “Reset” button. Continue flipping the coin, this time using
the “1000 flips” button. Complete the table below.
Flip #
Proportion of Heads
1000
.49
2000
.48
3000
.49
4000
.48
5000
.49
5. What seems to be happening?
With every 1,000 flips, the proportions of heads fluctuate by around .1 each.
6. Is this consistent with what you predicted in #3 above?
It is not consistent with what I predicted in #3
Let’s Roll the Dice
A coin only has two possible outcomes - “Heads” and “Tails”. So let’s try something a little more
complex. Let’s look at rolling a six-sided die, and let’s look at the probability of rolling a “six”.
1. If you roll a die one time, what are the possible proportions of rolling a “six”?
When you rolled a number other than 6, the proportion of rolling a 6 is 0.
When you rolled a number of 6, then the proportion of rolling a 6 is 1.
2. If you roll a die twice, what are the possible proportions of rolling a six?
The possible proportions of rolling a six 0/2, ½, and 2/2.
3. If you roll a die six times, what are the possible proportions of rolling a six?
The possible proportions of rolling a six is 0/6, ⅙, 2/6, 3/6, 4/6, ⅚, 6/6
4. If you roll a die a LOT of times, what do you predict the overall proportion of rolling sixes is?
I expect the overall proportion of rolling sixes is around 2/6 or 3/6.
Let’s simulate doing these die rolls. Open the Statistical Applet located at
https://www.statcrunch.com/applets/type3&dice. Leave it open in a separate browser window as you
work through this assignment together.
Allen Carandang
MATH 1090
5. Click on word “Outcomes” in the simulator. Then click on “1 roll” a few times. How many sixes
did you get?
I got 1 six.
6. Click on “reset” and then click on “Convergence”. Complete the table below by clicking on “1
roll” each time to find the cumulative proportion:
roll
#
Proportion
of Sixes
1
0
2
.5
3
.33
4
.5
5
.4
6
.33
7
.43
8
.38
9
.33
10
.3
11
.27
12
.25
13
.31
14
.36
15
.33
16
.31
17
.29
18
.28
19
.26
20
.3
7. What do you see happening?
With every dice rolled, the proportion of the sixes consists of a high of around .43 and a low of
Allen Carandang
MATH 1090
.26.
8. Reset the simulation by clicking the “Reset” button. Continue rolling the die, this time using the
“1000 rolls” button. Complete the table below.
Roll #
Proportion of Sixes
1000
.173
2000
.173
3000
.17
4000
.17
5000
.17
9. What seems to be happening?
With every 1,000 rolls, the proportion of sixes remains around a constant .17 proportion.
10. Is this consistent with what you predicted in #4 above?
It is not consistent with my predictions from #4.
Going from Proportions to Probability
Let’s remember the definition we put at the top:
The probability of any outcome of a random phenomenon is a number between 0 and 1 that
describes the proportion of times the outcome would occur in a very long series of repetitions.
So by this definition. The proportion of heads in a long run of coin tosses that we described in the “Let’s
Flip a Coin” section is also DEFINED to be the probability of getting a “head” on any given one time
coin flip. Similarly, the proportion of “sixes” in a long run of die rolls is DEFINED to be the probability
of getting a “six” on any given one die roll.
So, who cares? Unless you want to make a living as a professional gambler, or just randomly spend time
rolling dice and flipping coins, what in the world does any of this have to do with samples?