Statistical Fallacies and Probability Teasers

Under construction

Stock Trading – Incorrect Probability Conclusion

Stock can either go up , down or stay still. Hence, probability of any one stock going down is 1/3 rd.

Soldiers and Shell Bunkers – Incorrect Assumption

WW 1 soldiers were encouraged to get into fresh bunker shell holes – since it was deemed improbable that a shell would strike the same place twice. Incorrect- since , once the first event has occurred , it’s probability is 1 ! So , it doesn’t reduce from the joint probability of two events . Example- ace drawn from deck , then placed back . What is the probable that it will be drawn again? 1/52 ( not 1/52 times 1/ 52 cause first event has happened already)

The birthday problem

You need only 70 people in a room to have a 99.9 % chance of two common birthdays. You need just 23 for a 50% chance.

Are some runs more probable than others? (Is Chance lumpy  ?)

That is, if you flip a coin 1000 times, will you get more runs of 5 Heads than one would normally expect?

Simpson’s paradox – different sized groups

You solved 9 out of 10 problems, your friend solved 1 out of 1. His success rate was 100% yours was 90%. On day 2, you solved 0 out 1 and he solved 1 out of 2 – his success rate was 50%. Each day, his success rate was higher than yours.  This leads most people to believe that he had the better success rate overall (over the two days). However, over 2 days, you solved 9 out of 11 problems (82 % success), whereas he had 2 out of 3 (67 %). Thus, despite your friend solving a higher proportion of problems on each day, you actually won the challenge !

What if these were Clinical Trials instead of exam problems ? Important to understand the combined effect of multiple days.

The prosecutor’s fallacy

Confusion between two concepts –

• the probability of an individual matching the description of the perpetrator
• the probability of an individual (matching the description) of being guilty

Probability of an accused with the same description  – 1 in half a million ( 0.000002 ). 

When you bear in mind that the total population being dealt with is 10 million, 1 in half a million easily translates into 20 possible suspects.

The accused is one of 20 possible suspects – and the chance of his innocence is 19 in 20, not 1 in half-a-million. At least until they have an alibi for the remaining 19 people for that day and time.