How Not To Be Wrong

How Not to be Wrong by Jordan Ellenberg There are three lessons in the book: Lesson 1: Mathematics is mostly based on common sense, and we use it more than we think. For example, in WWII, military advisors looked at all the planes that returned from Europe, covered in bullet holes. Because the fuselage often …

Simpson’s Paradox

There are two drugs on the market for testing. Drug A cures 63 out of 90 people on day 1 while Drug B cures 8 out 10 people. On day 2, Drug A cures 4 of 10 people and Drug B cures 45 out of 90 people. On day 1, the recovery rate for A …

Josephus Problem

People are standing in a circle waiting to be executed. Counting begins at a specified point in the circle and proceeds around the circle in a specified direction. After a specified number of people are skipped, the next person is executed. The procedure is repeated with the remaining people, starting with the next person, going in the …

Probability

Flip a coin 5 times. Answer: it is looking for the total combination of k heads (order doesn't matter) over the total number arrangement outcome of flipping the coin n times. What is the probability that a head appears 3 times? Total combination of 3 heads = 5! / (3!*2!) = 10 Total combination of …

Permutation vs Combination

Permutation: order matters There are 7 people trying to sit on 3 chairs. How many sets of permutation are there? P (7, 3) = 7 * 6 * 5 = 210 P (n, k) = n! / (n - k)! Combination: order does NOT matter There are 7 people and we have to select 3 …

Three Prisoners problem

Three prisoners, A, B and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others …

Euler’s Number, E

Application: if you have a dollar and you invest it in a bank that pays 100% interest annually 1/12 interest monthly 1/52 interest weekly 1/365 interest daily What will be your investment worth at the end of the year 1? 1 * (1+100%)^1 = 2 1 * (1+1/12)^12 = 2.61 1 * (1+1/52)^52 = 2.69 …

Always Win in Soccer Betting

In China, the World Cup betting odds on the finalists are as follows: German wins at 1.24 Drawn at 4.8 Mexico wins at 8.7 If you are expecting to receive $100, you will need to invest as follows: German $100/1.24 = $80.65 Drawn $100/4.8 = $20.83 Mexico $100/8.7 = $11.5 The total is $113. In …

Secretary Problem

Imagine an administrator wants to hire the best secretary out of n applicants for a position. The applicants are interviewed one by one. A decision of "hired" or "rejected" must be drawn immediately after the interview. Once rejected, an applicant cannot be recalled. How should the administrator select the best candidate? The problem here is …

Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... This sequence of numbers appears in biological settings: branching in trees flower petals apple seeds spiral rows in sunflowers Plants do this because it is the most efficient way of packing things together in a small space. Using the number divided by …