I have a probability question that I need help with please. What is the probability that a number formed out of the ten numbers ranging 0-9 is an even number greater than 6 billion?

Question

directrix · Answer

Can digits be repeated?

anonymous · Answer

No

anonymous · Answer

the number is a ten-digit number with each of those numbers in it

directrix · Answer

I'm going to suggest an answer and ask you to examine it and see if you agree.

The number must be greater than 6 billion. Of the 10 digits, the only two digits that can begin the number are 6 and 8.

There are 10 "slots" (digits) in the number. In the first slot, goes a 2 for the two ways to begin the number.

-- TWO  --   --   --   --   --   --   --   --  --  --

directrix · Answer

That leaves 9 digits. An even digit has to be in the tenth slot to make the entire number even. There were 5 even digits (0,2,4,6,8) but one of those (the 6 or 8) had to begin the number. Now, there are 4 choices for the last digit.

-- TWO  --   --   --   --   --   --   --   --  -- FOUR --

anonymous · Answer

but also 7 and 9 can begin the number

directrix · Answer

At this point, there are 8 digits left for the remaining 8 slots of the number. For the second digit, there are 8 choices, for the third, there are 7 choices, and so on to the next to the last digit for which there will be 1 choice.

Multiply those and the number of possibilities for an even 10-digit number with value greater than 6 billion should be determined.

- TWO --EIGHT  -- SEVEN -- SIX --FIVE -- FOUR -- THREE -- TWO -- ONE -- FOUR 

That multiplies to 322, 560 number of ways.

Please check.

directrix · Answer

Okay. The number of choices for the first digit should be changed. What would you put there? And, what would be place as the last digit?

anonymous · Answer

Would you have to do two different probabilities because 2 chances of odd then 5 chances of even at the end or 2 chances even then 4 chances even?

directrix · Answer

I suppose the number of favorable outcomes could be calculated that way. What did you get?

directrix · Answer

The site went down. I cranked out the problem using two cases and wondered if you had had time to do that?