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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.

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?

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 ...