Download Thinking about probability…

Thinking About Chance
What are the chances?
What’s the chance of getting killed by lightning? How could we come up with a number?
US population = about 300 million, with an average of 80 people per year killed by lightning.
Chance of being killed by lightning:
What are the chances a randomly selected undergraduate student at the Columbia campus is
in the College of Arts and Sciences? Let’s look at enrollment data….
http://ipr.sc.edu/enrollment/prel2013/fall/fall_schools_official.htm
Thinking about probability…
Is the outcome of the coin toss before a football game known?
Why is it OK to use a coin toss to decide who get the ball first?
The idea behind probability…..
Chance behavior is unpredictable in the short run but has a regular and predictable pattern in
the long run.
Definitions
random – when individual outcomes are uncertain but there is a regular distribution of
outcomes in a large number of repetitions
probability – a number between 0 and 1 that describes the proportion of times an outcome
would occur in a very long series of repetitions
Random in statistics does not mean “haphazard”. In statistics, random describes a kind of
order that emerges only in the long run. Probability describes the long-term regularity of
events. Probability describes what happens in very many trials.
• probability = 0  the outcome never occurs
• probability = 1  the outcome happens on every repetition
• probability = ½  the outcome happens half the time in a very long series of
trials
How do we figure up the probability of something happening?
1. Empirical (or experimental) – repeat an experiment many times and calculate
proportion of time each outcome occurs.
2. Theoretical – if we make some assumptions based on a set of theories, we can calculate
probability based on this set of theories.
3. Personal Probability – this is our own personal judgment (not necessarily based on any
set of theories or previous observations)
Personal probabilities are not limited to repeatable settings. They’re useful because we
base decisions on them, this is simply expressing individual opinion. It can’t be said to be
right or wrong.
Let’s compute some probabilities and identify the type:
What is the probability of rolling a 2 with a fair (six-sided) die?
A. 1/6
B. 2/6
C. 3/6
What type of probability did you just compute?
A. Empirical
B. Theoretical
C. Personal
Let’s roll a die and observe the proportion of 2’s that come up.
What type of probability did you just compute?
A. Empirical
B. Theoretical
C. Personal
Suppose you have an idea of the probability of your favorite team winning this weekend.
Think about that probability. What type of probability did you just compute?
A. Empirical
B. Theoretical
C. Personal
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We can think about probability as the number of ways an outcome can happen divided by the
total number of possible outcomes.
# 𝑤𝑎𝑦𝑠 𝑓𝑜𝑟 𝑎𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑡𝑜 𝑜𝑐𝑐𝑢𝑟
𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 =
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Using the above formula, let’s re-examine the probability of rolling a 2 on a fair sided die.
Now use this formula to compute the probability of rolling an even number on a fair six sided
die?
A. 1/6
B. 2/6
C. 3/6
Now, let’s compute the probability of rolling a sum of 7 when rolling a fair die twice (or rolling
two fair dice at the same time).
How many possible outcomes when rolling a fair die twice?
How many possible outcomes are there when rolling 3 dice?
A. 6 + 6 + 6 = 18
B. 6 x 3 = 18
C. 6 x 6 = 36
D. 6 x 6 x 6 = 216
What is the probability of rolling a sum of seven with two fair dice? Recall that there are 36
possible outcomes.
A. 1/36
B. 6/36
C. 7/36
D. 14/36
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Law of Large Numbers (LLN) If a random phenomenon with numerical outcomes is
repeated many times independently, the mean of the observed outcomes (sample
mean) approaches the expected value (population mean).
In a large number of “independent” repetitions of a random phenomenon (such as coin
tossing), averages (or proportions, which really are averages) are likely to become more
stable as the number of trials increases.
The Law of Large Numbers is a statement about means and proportions, not sums and
counts.
The following picture is an example of the LLN in action
Question: Suppose you toss a fair coin 9 times and get “Tails” all 9 times. What are the
chances you’ll get another “Tails” on the 10th flip?
A. Highly Likely B. Not likely at all C. 50/50
Risk – Do we think about risk differently than the experts?
Think about cancer clusters. If you hear of a large number of people in a town that come
down with cancer, it can cause panic and fear. When we suspect a cluster in our
neighborhood, we tend to be upset and not think statistically. Let’s think about this.
Cancer accounts for more than 23% of all deaths in the US. So there are approximately
558,040 people that will die of cancer this year. That is a fairly high percentage, so there are
bound to be “clusters” of cancer cases just by chance. This is well studied.
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We have seen that sometimes the public does not think statistically….let’s explore this further.
We have two types of probability.
One looks at “personal judgment of how likely” –
The other looks at “what happens in many repetitions” –
Experts tend to look at “what happens in many repetitions”, and the public looks to “personal
judgment”. So, what’s considered risky?
Approximately 115 people die in an automobile accident every day in the US – that’s one
person every 13 minutes. Yet, we all get into cars on a regular basis…
Approximately 1440 college students will die this year due to alcohol related accidents
Yet, there are plenty of college students attending parties with alcohol on any given night…
If 1440 college students will die this year due to alcohol related accidents – this is
approximately 4 deaths per day
Why are there still plenty of parties on any given night where college students are drinking?
According to the US department of Education, roughly 20,000,000 students are enrolled in
college. What is the probability of a college student dying in an alcohol related accident this
year?
Why are there still plenty of parties on any given night where college students are drinking?
Why is there such a difference between what we consider risky and what the experts consider
risky?
1. We feel safer when a risk seems under our control than when we can’t
control it.
2. It’s hard to comprehend small probabilities.
3. The probabilities for some risks are estimated by experts from complicated
statistical models.
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ODDS
What are the odds? Odds are related to, but different than probability. Odds range from 0 to
infinity, while probability can range from 0 to 1. Odds can be stated in two ways and we’ll
want to convert them to probabilities.
First way: Odds in terms of 2 numbers: A to B
Odds of A to B for an outcome means that the probability of that outcome is A / (A + B).
So, “odds of 5 to 1 for a team winning” is another way of saying the “probability of the team
winning is 5/6.”
Second way: Odds in terms of 1 number: Θ
Odds of Θ for a team winning means that the probability is Θ / (1 + Θ)
“Odds of 5 for a team winning” is another way of saying the “probability of the team winning is
5/6.”
If the odds are 18 to 2 for a team winning, then the probability of the team winning is:
estimated to be:
A) 2/16 = 0.125
B) 2/18 = 0.111
C) 2/20 = 0.100
D) 16/18 = 0.889
E) 18/20 = 0.900
If the odds are 9 that a team wins, then the probability of the team winning is:
A) 9/10 = 0.900
B) 9/18 = 0.5
C) 1/9 = 0.111
D) 8/9 = 0.889
Note: Probabilities are intuitive for us to understand – they give us the chances of something
happening. Odds are…..well just the odds. It is a different way to quantify the likelihood of
something happening.
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