Sharing Birthdays

I was introduced to this challenge as a high school student. It was one of the most exciting moments of my student life! So here's the challenge..

In a class of 40 students, what is the probability that 2 students share the same birthday?

Very less! Don't you think so? 40 students and 365 days in a year and their birthdays matching... very very remote. However, against our intuition the probability is as high as 0.89!

The calculation is really simple. The probability that the birthdays of 2 students matches is (the second student has one chance in 365)

Now the probability that the birthdays of the 2 students doesn't match is

The probability that a third student doesn't have the same birthday as the first 2 students is

The probability that the 3 students don't share the same birthday is

(the probability of the birthday of the second student doesn't match with the first student AND the probability of the birthday of the third student not matching with the first and second students – product of both probabilities).

On similar lines, the probability that no 2 students have the same birthday out of a class of 40 is

and hence, the probability that 2 students have the same birthday out of a class of 40 is the complementary

Thats how the 89% chance that 2 students of a class have the same birthday. Observe how the probability increases steeply. Note that the Y axis in the graph is a logarithmic scale.

Related reading - 'Math Chambers' by Alfred S Posamentier

The Game of Four 4s

Game of Four 4s

The very thought of the end of the first term examinations gave such a relief to Ram who was studying in 6^th standard. He could now spend the next 3 weeks at home and enjoy cricket with his friends at the nearby ground. So much to do and all of it starts tomorrow, Ram thought.

He had informed all his friends to assemble at the ground at 6 am. He was just hoping that this grandmother doesn't stop him from playing in the sun. The next day Ram woke up and dashed off to the ground. This really surprised his mother. 'Normally it takes a good 15 minutes to wake him to go to school', she thought.

To his surprise, only 3 of his friends turned up. 4 was too small a number to play a game of cricket! They then decided to take turns to bat and bowl. But who would start first? Everyone wanted to bat first. This turned out to be a loud argument which attracted a retired teacher who was on his morning walk.

'What is the problem beta?', he inquired. Ram explained that he picked the number 4 in the random pickings but he didn't like being 4^th in the batting order. He didn't want to be last! The teacher was silent for a moment. He said, 'I am not sure why feel that way but for me 4 is a great number. May be best.'

Surprised to hear that, the kids wanted to know why it was so. The teacher then explained a game which was all about 4s. A game which looked pretty simple and straight-forward at first but went on to become more and more interesting.. try it for yourself

Rules of the game

1. You are allowed to use four 4s to construct the numbers 0, 1, 2, 3, 4,...
2. You can use any of the four fundamental operations of addition, subtraction, multiplication and division. Thus you can show the following -
   
   
   
   
3. You can also use the square root
   
4. You can also use powers
   
5. You can also use decimals both regular and recurring
   
   
6. You can then use the factorial. In general, for any integer N, N! = 1*2*3*4*....*N
   4! = 1*2*3*4 = 24

Using the above operations on the number 4, used 4 and only 4 times, using no other symbols (you can use parentheses), how far can you go starting from 1?