Question

A palindrome is chosen at random from the list of all $6$-digit palindromes, with all entries equally likely to be chosen. (Recall that a palindrome is a positive integer that reads the same forward and backwards, such as $387783$. Note: Since these palindromes are being thought of as numbers, rather than simply strings of digits, it is to be understood that the leading digit, the digit in the hundred-thousands place, must be nonzero.) What is the expected value of the chosen number?

Answer 1

The expected value of the chosen number is 55000.

All of the palindromes are of the form abcdab or abcdcd , where ab is a two-digit palindrome.

There are 99 valid two-digit palindromes, so there are twice as many palindromes of the first form, for a total of 2⋅99=198 palindromes of this form.

There are also 9 valid three-digit palindromes, so there are twice as many palindromes of the second form, for a total of 2⋅9=18 palindromes of this form.

Thus, there are a total of 198+18= 216​ palindromes.

Now, the palindromes of the form abcdab range from 10,001 to 99,999, inclusive, and the palindromes of the form abcdcd range from 1000,001 to 999,999, inclusive.

Thus, the average six-digit palindrome is 10001+99999 /2 =55000. Therefore, the expected value of the chosen number is 55000.

Answer 2

Expected value 550,000

Explanation:

Calculation of the expected value of the chosen number

We should know that each of the digit of the number may likely be thought of what we called a random variable, in which the first digits and the last digits comes uniformly from [1,2,3,4,5,6,7,8,9] while the last digit can't be 0 because of what we called the palindrome condition).

Therefore each of these two digits will have an expected value of 5 while the other four digits will come uniformly from this digits which are [0,1,2,3,4,5,6,7,8,9] in which each of the digits will have an expected value of 4.5.

Thus Expected value is additive, which means we have to also take place the value into account.

Now let find the EXPECTED VALUE

The expected value will be :

(100,000+1)⋅(5)+(10,000+1,000+100+10)⋅(4.5)

Expected value=550,000

Therefore the Expected value will be 550,000