A mathematical challenges for those who love maths.
Few months back, I came across ICICI Bank’s “upto 100%” cash back offer. When you use their credit card for shopping, you get a six digit number and depending on the sum of these six digits, you’ll get certain percentage of money back.
This post is an attempt to interpret the mathematical probability of winning 100% cash back as well as other combinations. (I do not have a solution for this yet, expecting some help from my readers)
The offer works like this: You use your ICICI Credit card and spend more than Rs 2000, your transaction slip will contain a 6 digit authorization code and you’ve to add the digits in the number to get the total. Depending on the total you get you’ll get certain amount of cash back on your purchase. For example, if your 6 digits sum up to say 51, you’ll get 100% refund while if its say 29 (i.e., for any sum between 15 and 30), you’ll get 1% refund.
Refer image for exact range and cash back offered.
The official link for this offer is here
It’s obvious that most of the numbers sum up between 21 and 30 and probability of the sum exceeding 51 or being less than 5 is very very low.
I wanted to know the exact probability for each combination. I’ve not reached the exact solution but below is the approach:
For a 6 digit number, with each digit ranging between 0 and 9. there're 10^6 possibilities between 000000 and 999999, i.e. 1000000 possible combinations
When you sum the digits, the minimum is 0 (when all digits are 0) and maximum is 54 (when all digits are 9). If you’ve to get 100% cash back, the sum should be either of 51, 52, 53 or 54
The probability of getting sum of 54 is 1 in 1000000
The probability of getting 53 is 6 in 1000000 (When the digit is 999998 or 999989
or 999899 or 998999 or 989999 or 899999)
The probability of getting 52 is 18 out of 1000000
(999997,999979,999799,997999,979999,799999,899998,899989,899899,898999,889999,989998,
989989,989899,988999,998998,998989,998899)
The chance of getting 51 is 58 out of 1000000 ( 999996,999969,999699,996999,969999,699999,799998,799989,799899,798999,789999,979998,
979989,979899,978999,997998,997989,997899,999798,999978,899997,899979,899799,897999,
879999,989997,989979,989799,987999,998997,998979,998799,999897,999987,888999,988899,
998889,999888,,899988,899889,898899,889998,889989,889899,898998,898989,898899,
899898,899889,899988,989988,989889,988998,988989,989898,989889,989988,998988)
This means, the total probability of getting the sum between 51 and 54 is 1+6+18+58=83 out of 1000000, which is 0.000083
In other words, your chance of getting 100% cash back is 0.0083 % (8 per thousand attempts) which means, if you spend Rs 2,000 or above One thousand time, chances are that eight times you'll get back the money you spent (Rs 16,000 out of 20,00,000 Rs you spend.
Note: Thanks to Surya for correcting me in maths. I'd done some wrong calculations earlier. His profile is not accessible hence I've not given a link, details in comment section.
Similarly, the probability of the sum falling less than 5 is also very very low.
The probability is maximum for 21-30 range, hence a minimum cash back is offered there.
Note: I’ve done only a manual calculation and there can be errors. There should be a better mathematical formula to calculate this specific case study. The conventional permutation and combination formulas I know (nPr and nCr) can't be applied in this case. I consulted many of my friends and staff for a solution but either because they were busy or couldn’t find a solution; I don’t have a mathematical formula to calculate the probability for all combinations. If you can help, it’ll be great.
Since this offer is implemented commercially nationwide, whoever designed this should have a done his ground work carefully because had that person done a wrong calculation, ICICI would have lost crores of rupees in cash back. I salute the brain that designed this scheme.
Here’s an open mathematical challenge:
Can you come up with the mathematical formula for this offer?
What is the probability that I'll get 75% cash back? And for other percentages as well? Will be great if you can help.
Note: This blogger assumes that Mr. K. Vaman Kamath and his ICICI Bank are honest in their approach and no cheap tricks are adopted in this offer. If they’re determined not to give 75% or 100% cash back to anyone, all it takes is few extra lines of software code that will prevent generation of authorization codes, digits of which do not sum up to specific numbers for which huge cash backs are offered.
The scheme may appear like yet another marketing strategy for credit cards but the mathematical calculation involved deserves an appreciation. The bank obviously can't afford to go bankrupt by giving huge cash backs to all, but there's no harm trying your luck. (1% refund is guaranteed anyway)
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