Will they ever voluntarily help out?
What I'm asking is "will they tend to choose activities other than play at some point". And I mean activities traditionally thought of as work,
Is putting all the video game discs into a CD wallet work or play?
Is noticing the sink is dirty and cleaning it work or play? What if it isn't your sink or your responsibility?
Is making your own bed knowing that it could be left unmade for the rest of your life work or play?
The division of the tasks around the house should be between "worth my time" and "not worth my time" rather than play and work.
There really isn't a good answer to your question. It depends on the atmosphere of the house rather than age. Kids will help out because it feels good to do it. Our job is to create an atmosphere so they can feel good about helping, or an atmosphere that doesn't crush that feeling ... so that "work" feels good.
Someone was asked how they got their child to like broccoli. She answered, "I didn't do anything to make her dislike broccoli." That goes for everything. :-) Broccoli, writing, household tasks, astronomy, reading and so on. Don't do anything to make them dislike helping you.
When I said "grow up" about my kids I was wondering when in their lives I would begin to see that tendency in them to understand that part of our lives is "necessary application" and other is fun and discovery.
At 11 my daughter Kat began voluntarily helping me to do things like bring in the groceries and put them away. A couple of times she noticed the bathroom sink was dirty and cleaned it and then asked what I thought of the job she did to make sure I'd noticed she'd done it. :-) She's 12 now and has seen the cat litter scattered on the floor of the bathroom a number of times and has vacuumed it up without even mentioning she did it.
She's doing these things not because she sees the necessity but because she wants to. There's a part of her world that affects her sensibilities of rightness and she recognizes she has the ability to right it to her standards. It's part of changing body chemistry which changes the way humans see their world.
As she was growing up I've done both demanding she help -- when company is coming and I'm in a tizzy -- and I've asked if she'd like to help. Asking works much better!
Demanding gets more work but it creates a negative relationship and causes her to not want to volunteer. I realized that getting grumpy and making her cry was too great a price to pay for something that not only wouldn't make a difference in 10 years but, in fact, I was making her upset for a mess that would very likely return tomorrow.
When Kat was younger, housework was well beyond her. Even low key things like folding kitchen towels or sorting socks or underwear, she had no interest in. Given the choice of helping so it goes faster or waiting for me to finish, she always chose waiting (impatiently ;-). So asking her didn't get me much help, but when she did want to do something, like the stirring and pouring parts of cooking, the time was spent happily together.
My mother was made to do chores as a kid and she decided she'd never do that to her kids. And she didn't. She did all the work. She also assumed that at some point my sister and I would help out. But we didn't. I think it was because she never invited us along to help. Then it puzzled my sister and me as teens why she'd be upset that we weren't helping when she'd never needed help before.
I think kids can be trained early to do things parents want them to do. They can also be trained not to complain. Sometimes at puberty kids then get that urge to set their environment right and the parents feel like they did a good job training them to do chores. But sometimes at puberty kids realize they have the power to demand that they not be pushed around anymore and they suddenly seem to turn into "bad kids" when the parents felt they were doing everything right all along.
So bottom line is to shift your point of view to recognizing that the house and the decisions about how it's maintained are yours. Recognize that you have a choice about how it's kept. No one makes you clean to a certain standard. Give yourself permission not to do things. It helps a great deal when you shift your perspective from having to do the dishes to doing them because you would like to have some clean plates.
And then, invite the kids along to help. Create an atmosphere that they'd like to join in. (Hint: being a grump because company's coming in 2 hours and the house is a disaster is not a good way ;-) Rather than presenting the invitations as soliciting help with something burdensome, present the invitation as an opportunity to spend time with you and the work is incidental. You can even just ask if they'd like to do something in the same room with you while you do work on something or bring the work to them, like folding laundry, so you can be in the same room they're playing in.