Why does the author's answer for (b) and (c) differ from my answer of 1-0.1k?

[WMDhamnekar]

Homework Statement

What is the probability that among k random digits (a) 0 does not appear; (b) 1 does not appear; (c)neither 0 nor 1 appears; (d) at least one of the two digits 0 and 1 does not appear ? Let A and B represents the event in (a) and (b). Express the other events in terms of A and B.

Relevant Equations

No relevant equations

(a) The probability that 0 appears k times in k random digits is 0.1^k So, It does not appear in k random digits is 1 - 0.1^k. But author says 0.9 ^k.
How is that?

(b) My answer is same as in (a) that is 1-0.1^k. Author's answer is 0.9^k.

(c)1 - 0.1^k - 0.1^k Author's answer is 0.8^k. How is that?

How to answer (d) and other remaining part of the question?

 

[Doc Al]

(a) The probability that 0 appears k times in k random digits is 0.1k So, It does not appear in k random digits is 1 - 0.1k.

But what about appearing k-1 times? And k-2 times? If you found the probability for any appearance, then you can subtract from one to find the probability for no appearance.

But author says 0.9 k.

(probability of not appearing in slot 1)*(probability of not appearing in slot 2)... = (.9)*(.9)...

 

[PeroK]

Homework Statement:: What is the probability that among k random digits (a) 0 does not appear; (b) 1 does not appear; (c)neither 0 nor 1 appears; (d) at least one of the two digits 0 and 1 does not appear ? Let A and B represents the event in (a) and (b). Express the other events in terms of A and B.
Relevant Equations:: No relevant equations

(a) The probability that 0 appears k times in k random digits is 0.1^k So, It does not appear in k random digits is 1 - 0.1^k. But author says 0.9 ^k.
How is that?

(b) My answer is same as in (a) that is 1-0.1^k. Author's answer is 0.9^k.

(c)1 - 0.1^k - 0.1^k Author's answer is 0.8^k. How is that?

How to answer (d) and other remaining part of the question?

If you use the digits $0, 1, 2$ only to simplify things and take $k = 3$, say, then you should be able to simulate the experiment and understand what is happening. E.g. For a) you have:

111
112
121
122
211
212
221
222

That have no $0$.

Once you understand this example, you can extend it to 0-9 and then to $k$ digits for any $k$.

 

[FactChecker]

(a) The probability that 0 appears k times in k random digits is 0.1^k So, It does not appear in k random digits is 1 - 0.1^k. But author says 0.9 ^k.
How is that?

$0.1^k$ is the probability that 0 appears in all $k$ digits. It's the probability of $0_1 0_2 ... 0_k$.
As a sanity check, notice that as $k$ gets larger, your answer, $1-0.1^k$ approaches 1. Does that make sense? That implies that as $k$ gets huge, it becomes almost certain that no $0$ appears. In reality, as $k$ gets huge, say a million, it is almost certain that there will be at least one zero.

The correct answer is that the other digits DO appear in all $k$ places. That probability is $0.9^k$.

Your answers to the other parts must be adjusted accordingly.