1. How small probabilities affect our life?
By: Boon Xuan, Fatin & Mei Ying (Group 1)
Based on: Leonard Mlodinow’s The drunkard’s walk: how
randomness rules our lives. (2008) - Chapter 2

2. What do people usually think about probabilities?
Also known as ‘chance’
BUT IT’S MUCH MORE THAN THAT
Textbook examples:
-Rolling a dice
-Flipping a coin
Common real world applications:
-Weather prediction
-Winning a lottery

3. ~Imaginary scenario~
-You, a totally innocent person, was accused of a murder that you happened to witness from afar.
-A DNA sample was taken from you so as to check if it matches the perpetrator's
-Results came back positive that your DNA matches the murderer’s!!

4. You Jury How could this happen?! Maybe it is a false positive??
Jury
There is a 1 in 1 billion accidental match & 1 in 100 lab-error match. So the overall probability of a false positive is 1 in 500 million.
Your argument is invalid!! You are guilty!!

5. How small probabilities affect our life?- the legal system
Wrong use of probabilities can lead to disastrous consequences:
DNA test example
“1 in 1 billion accidental match & 1 in 100 lab-error match- so the overall probability of a false positive is 1 in 500 million”
What it should have been:
1 / 1 billion + 1/100 = ~ 1/100 probability of false positive ✔
Although 1/100 is a more accurate probability, this probability is not recognised in the legal system and therefore courts tend to believe that it is almost impossible to get false positives in DNA testing!

6. How small probabilities affect our life?- the legal system
Had the court taken into account the actual error probability in DNA testing, such cases could have been avoided.
Small probabilities matter!
Wrong use of probabilities can lead to disastrous consequences:
Real life example:
Error in DNA analysis by the lab led to Timothy Durham being wrongly accused of rape. A DNA re-test showed that the perpetrator's DNA did not match Durham’s.
He was later exonerated from his charges, but had already spent almost 4 years in prison.
Timothy Durham (Photo/Innocence Project)
Small probabilities matter because it can affect our fate.

7. Rule for compounding probabilities:
Small probabilities might lead to wrong conclusions when basic laws of probability are not applied
Rule for compounding probabilities:
If two possible events, A and B, are independent, then the probability
that both A and B will occur is equal to the product of their
individual probabilities.

8. The probability of both happening is 1/25000
Probability of getting married to a policeman in a year is 1/50
Probability of a policeman dying on his job in a year is 1/5000
The probability of both happening is 1/25000

9. However, the cases mentioned are not independent because once the policeman dies, he can’t get married
Hence, this statistic is not accurate and such inaccuracies might lead us into making the wrong decision
i.e.

10. How small probabilities affect our life?- the legal system
Second Scenario of using probability in an incorrect manner:
Characteristic
Individual Probability
Partly yellow automobile
1/10
Man with mustache
1/4
Negro man with beard
Girl with ponytail
Girl with blond hair
1/3
Interracial couple in car
1/1000

11. How small probabilities affect our life?- the legal system
To use the product rule of probability, the categories have to be independent, but the problem is that they are not independent. According to the table, the chance of observing a “Negro man with beard” is 1 in 10 while a “man with mustache” is 1 in 4. However a man with a beard usually also have a mustache. Therefore the chances of observing a “Negro man with beard” and a “ man with mustache” is not 1/10 x 1/4, it should be higher than that!
Small probabilities matter!
Second Scenario of using probability in an incorrect manner:
Error
Product rule applies to the data → Chances of the couple being innocent is 1 in 12 million

12. How small probabilities affect our life?- the legal system
The correct probability should be: the chance that a couple who fits all the descriptions is the guilty couple. We also have to consider the population of the area adjoining the crime scene which was several millions, so logically there should be about 2 or 3 couples in the area who matched the description. Therefore the probability that a couple who matched the description was guilty is only 1 in 2 or 3 if we purely based it on this calculation.
Small probabilities matter!
Second Scenario of using probability in an incorrect manner:
Error
the relevant probability is the one stated above—the probability that a couple selected at random will match the suspects’ description

13. In conclusion..
Probability problems may be straight-forward in textbooks but in real life they are not always used correctly.
People tend to ignore analysing small probabilities because they seem rather insignificant.
However these oversights may result in the wrong conclusion obtained→ possibly depriving us of a happy ending.
Remember guys, probability is important! ;)