Question: 4-Digit Number

Comment on 4-Digit Number

don't you think the 2nd, 3rd and 4th digits should have a probability of 1/9th since the 1st digit cannot be 0 so the following digits cannot be 0 either ? How do we go about this then ?
greenlight-admin's picture

No, it's still 1/10 (although it certainly seems like it might be 1/9).
However, it doesn't matter that the 1st value can't be 0. There are 10 possible values for the 2nd digit, and only 1 of them matches the 1st digit. So, the probability that the 2nd number matches the first is 1/10

For probability approach: Why is the first digit not 9 out of 10 since 0 is not allowed? So isn't 0 not being allowed a restriction for the first digit?
greenlight-admin's picture

We're selecting an integer from 1000 to 9999, so it's impossible for the 1st digit (the thousands digit) to be zero.

Can we imply that the selection of the 2nd to 4th digits is equivalent to "selection with replacement"?
greenlight-admin's picture

Yes, that would be an equivalent model.
Great idea!

Why isn't the probability of 1st digit 1/9?
greenlight-admin's picture

The first digit can be ANY digit among 1, 2, 3, 4, 5, 6, 7, 8, and 9

It doesn't matter which digit it is. What matters is that the remaining digits match the first digit.

Does that help?

Why is the probability of the first digit just 1
greenlight-admin's picture

If we want all 4 digits to be the same, then it doesn't matter what the first digit is.

For example, if the first digit is a 7 then, in order for all 4 digits to be the same, we need the other three digits to be 7.

Similarly, if the first digit is a 9 then, in order for all 4 digits to be the same, we need the other three digits to be 9.

If the first digit is a 2 then, in order for all 4 digits to be the same, we need the other three digits to be 2.

So, the likelihood of all 4 digits being the same does not depend on the value of the first digit.
In fact, the first digit can be ANY digit.

If we randomly choose a digit to be the first digit, what is the probability that the selected digit is ANY digit?

Since our first selection is guaranteed to be an integer, the probability is 1.

Does that help?


I have this question in mind >) How come selecting the 4 digits constitutes as independent events when we clearly see that they should match the previous digits?

greenlight-admin's picture

Each digit is RANDOMLY selected.

So, for example, if we select 7 for the first digit, this has no affect on the likelihood of randomly selecting a 7 for the second digit.
So, those two events are independent.
We can apply the same logic to show that all 4 events (selecting the 4 digits) are independent.

Does that help?


This definitely helps. Thanks!

My approach:
Total outcomes = 9000
favourable outcomes: 9 (1111,2222,3333 and so on up to 9999)
hence probability = 9/9000 = 1/1000
greenlight-admin's picture

That's a perfect approach (it's also the same as the first solution in the video :-)

Is the following a valid approach: Solve for the denominator using y-x+1 (as shown in the video, 9999-1000+1). Then use the Fundamental Counting Principal to determine the numerator (Breaking the task into 4 stages with stages 2,3, & 4 being the most restrictive: 9x1x1x1)? Giving you 9/9000.
greenlight-admin's picture

Great approach. Nice work!

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