## Solutions to Birthday Problems: IITJEE Advanced Mathematics

In the following problems, each year is assumed to be consisting of 365 days (no leap year):

Problem 1:

What is the least number of people in a room such that it is more likely than not that at least two people will share the same birthday?

Solution 1:

The probability of the second person having a different birthday from the first person is $\frac{364}{365}$. The probability of the first three persons having different birthdays is $\frac{364}{365} \times \frac{363}{365}$. In this way, the probability of all n persons in a room having different birthdays is $P(n) = \frac{364}{365} \times \frac{363}{365} \times \frac{362}{365} \times \ldots \frac{365-n+1}{365}$. For the value of n, when P(n) falls just below 1/2 is the least number of people in a room when the probability of at least two people having the same birthday becomes greater than one half (that is, more likely than not). Now, one can make the following table:

$\begin{tabular}{|c|c|}\hline N & P(n) \\ \hline 2 & 364/365 \\ \hline 3 & 0.9918 \\ \hline 4 & 0.9836 \\ \hline 5 & 0.9729 \\ \hline 6 & 0.9595 \\ \hline 7 & 0.9438 \\ \hline 8 & 0.9257 \\ \hline 9 & 0.9054 \\ \hline 10 & 0.8830 \\ \hline 11 & 0.8589 \\ \hline 12 & 0.8330 \\ \hline 13 & 0.8056 \\ \hline 14 & 0.7769 \\ \hline 15 & 0.7471 \\ \hline 16 & 0.7164 \\ \hline 17 & 0.6850 \\ \hline 18 &0.6531 \\ \hline 19 & 0.6209 \\ \hline 20 & 0.5886 \\ \hline 21 & 0.5563 \\ \hline 22 & 0.5258 \\ \hline 23 & 0.4956 \\ \hline \end{tabular}$

Thus, the answer is 23. One may say that during a football match with one referee, it is more likely than not that at least two people on the field have the same birthday! 🙂 🙂 🙂

Problem 2:

You are in a conference. What is the least number of people in the conference (besides you) such that it is more likely than not that there is at least another person having the same birthday as yours?

Solution 2:

The probability of the first person having a different birthday from yours is $\frac{364}{365}$. Similarly, the probability of the first two persons not having the same birthday as yours is $\frac{(364)^{2}}{(365)^{2}}$. Thus, the probability of n persons not  having the same birthday as yours is $\frac{(364)^{n}}{(365)^{n}}$. When this value falls below 0.5, then it becomes more likely than not that at least another person has the same birthday as yours. So, the least value of n is obtained from $(\frac{364}{365})^{n}<\frac{1}{2}$. Taking log of both sides, we solve to get $n>252.65$. So, the least number of people required is 253.

Problem 3:

A theatre owner announces that the first person in the queue having the same birthday as the one who has already purchased a ticket will be given a free entry. Where (which position in the queue) should one stand to maximize the chance of earning a free entry?

Solution 3:

For the nth person to earn a free entry, first $(n-1)$ persons must have different birthdays and the nth person must have the same birthday as that of one of these previous $(n-1)$ persons. The probability of such an event can we written as

$P(n) = [\frac{364}{365} \times \frac{363}{365} \times \frac{362}{365} \times \ldots \frac{365-n+2}{365}] \times \frac{n-1}{365}$

For a maximum, we need $P(n) > P(n+1)$. Alternatively, $\frac{P(n)}{P(n+1)} >1$. Using this expression for P(n), we get $\frac{365}{365-n} \times \frac{n-1}{n} >1$. Or, $n^{2}-n-365>0$. For positive n, this inequality is satisfied first for some n between 19 and 20. So, the best place in the queue to get a free entry is the 20th position.

More later,

Nalin Pithwa.

## Three More Quickies

Reference:

“Professor Stewart’s Cabinet of Mathematical Curiosities”, Ian Stewart.

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1. If five dogs dig five holes in five days, how long does it take ten dogs to dig ten holes? Assume that they all dig at the same rate all the time and all holes are the same size.
2. A woman bought a parrot in a pet shop. The shop assistant, who always told the truth said, “I guarantee that this parrot will repeat every word it hears.” A week later, the woman took the parrot back, complaining that it hadn’t spoken a single word. “Did anyone talk to it?” asked the suspicious assistant. “Oh, yes.”. What is the explanation?
3. The planet Nff-Pff in the Anathema Galaxy is inhabited by precisely two sentient beings, Nff and Pff. Nff lives on a large continent, in the middle of which is an enormous lake. Pff lives on an island in the middle of the lake. Neither Nff nor Pff can swim, fly or teleport: their only form of transport is to walk on dry land. Yet, each morning, one walks to the other’s house for breakfast. Explain.

Shared by Nalin Pithwa.

## Birthday Probability Problems: IITJEE Advanced Mathematics

In the following problems, each year is assumed to be consisting of 365 days (no leap year):

1. What is the least number of people in a room such that it is more likely than not that at least two people will share the same birthday?
2. You are in a conference. What is the least number of people in the conference (besides you) such that it is more likely than not that there is at least another person having the same birthday as yours?
3. A theatre owner announces that the first person in the queue having the same birthday as the one who has already purchased a ticket will be given a free entry. Where (which position in the queue) should one stand to maximize the chance of earning a free entry?

I will put up the solutions on this blog tomorrow. First, you need to make a whole-hearted attempt.

Nalin Pithwa.

## Message for students: Ms. Sumita Mukherjee, Principal, Ryan International School, NOIDA

“You can lose everything, but not education” :

Ms. Sumita Mukherjee, Principal, Ryan International School, NOIDA, talked to DNA, (Mumbai, print edition, May 19 2017) about  the growing concern of peer pressure among young students, the importance of sex education in schools and more: Excerpts from the interview:

How can one help students struggling with peer pressure?

There are a lot of students dealing with performance and peer pressure. It’s very common among teenagers. First of all, we need to identify such students in our schools, and then understand their issues. Recently, we identified a Class 12 student in our school who was brilliant till Class 11. He suddenly stopped coming to school regularly. We called up his parents and what we got to know that the excuse he had given to them was that nothing important was happening at school. We asked the parents how they can take it for granted. We talked to him and realized that he was going through peer pressure. We told him about the challenges of life and convinced him that he can’t give up on his future like that. Now, he attends school regularly. So, we need to handle such students very carefully and it is also the responsibility of parents to approach the school immediately when something like this happens.

$\vdots$

Any message for students?

You can lose everything in life, but not education.

************************************************************************************************

Shared by Nalin Pithwa.

## JEE Score (will) still count for (admissions to) IIT’s

Reference: DNA newspaper, print edition, May 15 2017, Mumbai, Section: Education;

The All India Council for Technical Education (AICTE) which oversees aspects related to technical education in the country, had discussed at a meeting early this year a proposal for having a single entrance test for engineering colleges for undergraduate courses. Admission to Indian Institutes of Technology (IIT’s) will, however, continue to happen on the basis of Joint Entrance Examination (JEE) only.

## The animals went in which way

The animals may have gone into Noah’s Ark two by two, but in which order did they go in? Given the following sentence (yes, sentence! — I make no apologies for the punctuation), what was the order in which the animals entered the Ark?

The monkeys went in before the sheep, swans, chickens, peacocks, geese, penguins and spiders, but went in after the horses, badgers, squirrels and tigers, the latter of which went in before the horses, the penguins, the rabbits, the pigs, the donkeys, the snakes and the mice, but the mice went before the leopards, the leopards before the squirrels, the squirrels before the chickens, the chickens before the penguins, spiders, sheep, geese and the peacocks, the peacocks before the geese and the penguins, the penguins before the spiders and after the geese and the horses, the horses before the donkeys, the chickens and the leopards, the leopards after the foxes and the ducks, the ducks before the goats, swans, doves, foxes and badgers before the chickens, horses, squirrels and swans and after the lions, tigers, foxes, squirrels and ducks, the ducks after the lions, elephants, rabbits and otters, the otters before the elephants, tigers, chickens and beavers, the beavers after the elephants, the elephants before the lions, the lions before the tigers, the sheep before the peacocks, the swans before the chickens, the pigs before the snakes, the snakes before the foxes, the pigs after the rabbits, goats, tigers and doves, the doves before the chickens, horses, goats, donkeys and snakes, the snakes after the goats, and the donkeys before the mice and the squirrels.

🙂 🙂 🙂

Nalin Pithwa.

## The power of the unseen, the abstract: applications of mathematics

Applications of math are everywhere…anywhere we see, use, test/taste, touch, etc…

I have made a quick compilation of some such examples below:

1. Crystallography
2. Coding Theory (Error Correction) (the stuff like Hamming codes, parity check codes; used in 3G, 4G etc.) Used in data storage also. Bar codes, QR codes, etc.
3. Medicine: MRI, cancer detection, Tomography,etc.
4. Image processing: JPEG2000; Digital enhancement etc.
5. Regulating traffic: use of probability theory and queuing theory
6. Improving performance in sports
7. Betting and bidding; including spectrum auction using John Nash’s game theory.
8. Robotics
9. Space Exploration
10. Wireless communications including cellular telephony. (You can Google search this; for example, Fourier Series is used in Digital Signal Processing (DSP). Even some concepts of convergence of a series are necessary!) Actually, this is a digital communications systems and each component of this requires heavy use of mathematical machinery: as the information bearing signal is passed from source to sink, it under goes several steps one-by-one: like Source Coding, encryption (like AES, or RSA or ECC), Error Control Coding and Modulation/Transmission via physical channel. On the receiver or sink side, the “opposite” steps are carried out. This is generally taught in Electrical Engineering. You can Google search these things.
11. DNA Analysis
12. Exploring oceans (example, with unmanned underwater vehicles)
13. Packing (physical and electronic)
14. Aircraft designing
15. Pattern identification
16. Weather forecasting.
17. GPS also uses math. It uses physics also. Perhaps, just to satisfy your curiosity, GPS uses special relativity.
18. Computer Networks: of course, they use Queuing theory. Long back, the TCP/IP slow start algorithm was designed and developed by van Jacobson.(You can Google search all this — but the stuff is arcande right now due to your current education level.)
19. Architecture, of course, uses geometry. For example, Golden ratio.
20. Analyzing fluid flows.
21. Designing contact lenses for the eyes. Including coloured contact lenses to enhance beauty or for fashion.
22. Artificial Intelligence and Machine Intelligence.
23. Internet Security.
24. Astronomy, of course. Who can ever forget this? Get yourself a nice telescope and get hooked. You can also Stellarium.org freeware to learn to identify stars and planets, and constellations.
25. Analyzing chaos and fractals: the classic movie “Jurassic Park” was based on fractal geometry. The dino’s were, of course, simulations!
26. Forensics
27. Combinatorial optimization; the travelling salesman problem.
28. Computational Biology

We will try to look at bit deeper into these applications in later blogs. And, yes, before I forget “Ramanujan’s algorithm to compute $\pi$ up to a million digits is used to test the efficacy and efficiency of supercomputers. Of course, there will be other testing procedures also, for testing supercomputers.

There will be several more. Kindly share your views.

-Nalin Pithwa.