Maths x Life

I came across this riddle about aeroplanes and probability (two of my favourite topics!): There are 100 passengers lined up (in a random order) to board a plane. The plane is fully booked, meaning there are exactly 100 seats available, assigned to the 100 passengers. The first passenger chooses a seat at random, with all seats equally likely. Then, each of the other passengers proceeds as follows: if their assigned seat is free, they will sit in it; otherwise, they will take a random available seat. What is the probability that the last passenger will sit in their assigned seat? Here's my analysis (using the terminology P1-P100 for the passengers, S1-S100 for the assigned seats, and DP1-DPn for the “displaced” passengers): For passenger P1:  P1 (randomly) makes one of   three choices: Sits in his own seat (S1) , in which case everyone sits in their assigned seat.  Sits in the last passenger’s seat (S100) , in which case everyone else sits in their assigned seat, except P1...

I first saw this problem when I was watching the movie  ‘ 21’ . The re is a short classroom scene where Professor Micky Rosa tests Ben Campbell with what sounds like a simple probability puzzle.  This is their conversation. Professor: Ben, suppose you're on a game show and you are given a chance to choose from three different doors. Now behind one of the doors is a new car and behind the other two – goats. Which door would you choose?    Ben: Door number 1.    Professor: Ben chooses door number 1! The game show host, who by the way knows what's behind all the other doors, decides to open another door. Let's say he chooses door number 3… behind which sits a goat. Now, Ben – the game show host comes up to you and says, “Ben, do you want to stay with door number 1 or go with door number 2”? Now is it in your interest to switch your choice?   Ben: Yeah.   Professor: Well, wait – remember the host knows where the car is so how do you know he's not play...

This  story is about an  Indian restaurant in London called   ‘Dishoom’ ,  which has ingeniously applied probability theory and economics to its business model.    Dishoom found that while they are generally booked out, they tend to have vacant tables before 6pm, Monday to Thursday. So, they came up with a unique loyalty program in which members who pay their bill before 6pm, Monday to Thursday, are given a die to roll at the end of their meal; if a 6 is rolled, the meal is free. Apart from the maths (discussed below), the idea is clever because of its simplicity and the buzz that it has been able to create.   Here's the maths.     For the customer, there’s the excitement of the possible free meal and the luck of the roll – a 1 out of 6 chance that the meal will be free. However, the mathematical analysis is as follows. Assuming a meal cost of £60, the customer’s expected saving is (1/6 × 60) + (5/6 x 0) = £10 pe r the expected...

From the first time I saw a Pringles tube, I had wondered about why these chips are shaped the way they are, different from the “randomness” of other chips. The simple answer was that Pringles are not made from potato slices, but from extruded potato paste – like the snacks ‘sev’ and ‘chakli’. To be honest, I never thought too much about the shape again till I recently made the connection while revising slope fields and thought about the sketching of quadric surfaces (graphs of polynomials with the highest power of 2 in 3 dimensions). Having now learnt about the maths of this shape, a hyperbolic paraboloid, this story is about the design of the Pringles chip. The shape is called a hyperbolic paraboloid because the equation gives a parabola in two planes (xz and yz planes), and a hyperbola in one (xy plane). One way to think about this is by thinking of cross sections of the curve – like an MRI. If we take a horizontal slices (z = constant) of a Pringle, then we get hyperbolic sections ...