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This problem is a very easy and cute problem of probability from ISI MStat PSA 2019 Problem 18.

Draw one observation \(N\) at random from the set \(\{1,2, \ldots, 100\}\). What is the probability that the last digit of \(N^{2}\) is \(1\)?

- \(\frac{1}{20}\)
- \(\frac{1}{50}\)
- \(\frac{1}{10}\)
- \(\frac{1}{5}\)

Last Digit of Natural Numbers

Basic Probability Theory

Combinatorics

But try the problem first...

Answer: is \(\frac{1}{5}\)

Source

Suggested Reading

ISI MStat 2019 PSA Problem Number 18

A First Course in Probability by Sheldon Ross

First hint

Try to formulate the sample space. Observe that the sample space is not dependent on the number itself rather only on the last digits of the number \(N\).

Also, observe that the number of integers in \(\{1,2, \ldots, 100\}\) is uniformly distributed over the last digits. So the sample space can be taken as \(\{0,1,2, \ldots, 9\}\). So, the number of elements in the sample space is \(10\).

**See the Food for Thought**!

Second Hint

This step is easy.

Find out the cases for which \(N^2\) gives 1 as the last digit. Use the reduced last digit sample space.

- 1 x 1
~~3 x 7~~(Since \(N^2\) and they must have the same last digit)~~7 x 3~~(Since \(N^2\) and they must have the same last digit)- 9 x 9

So, there are 2 possible cases out of 10.

Third Step

Therefore the probability = \( \frac{2}{10} = \frac{1}{5}\).

Food For Thought

- Observe that there is a little bit of handwaving in the First Step. Please make it more precise using the ideas of Probability that it is okay to use the sample space as the reduced version rather than \(\{1,2, \ldots, 100\}\).

- Generalize the problem for \(\{1,2, \ldots, n\}\).

- Generalize the problems for \(N^k\) for selecting an observation from \(\{1,2, \ldots, n\}\).

- Generalize the problems for \(N^k\) for selecting an observation from \(\{1,2, \ldots, n\}\) for each of the digits from \(\{0,1,2, \ldots, 9\}\).

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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