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Three probability problems

4. What are the odds that a randomly selected number between 1 and 100 will not have a digit 7?
(For instance, 23 does not have a digit 7, but 73 does.)

5. There are 4 slots, each containing any of letters a, m, h, t.

HINT:
What are the odds that letters stored in these slots read the word math?
Remember that each slot may contain any of the 4 letters. For this problem "aaaa" is a possibility.

6. Consider strings of length 9 with elements being a, b or c. How many strings will contain at least 6 bâ??s?

HINT:
This one is probably the most difficult problem on the homework. Try to break up the event of "AT LEAST 6 bâ??s" into smaller mutually exclusive events that are easier to work with. For example, consider first finding the number of ways that a string could contain EXACTLY 6 bâ??s. Then try EXACTLY 7 bâ??s...
Once you found how many ways to have put in the 6 b's, you are not done with the exactly 6 part. You still need to account for the remaining 3 spaces, each of which can be filled with an "a" or "c" (otherwise it wouldn't have 6 b's).

Solution Preview

4. The answer to this question depends on whether 1 and 100 are included in the selected numbers, so I will give you the answers for the difference configurations:
If the number range is from 1 to 99 inclusively, then there are 99 numbers total and 80 do not contain a 7, so 19 do contain a 7. So the odds that a randomly selected number will not ...

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