I definitely think you can get the intuitions behind probability to elementary age kids, and doing so might even be a way to introduce the important notions of equivalent fractions / related rates / etc.
Take a six-sided die and mark 1 on one side, 2 on two sides, 3 on three sides. Show the kids the die. Ask them, if you roll this die a bunch of times, which number do you think will come up most often? Ask them how much more often they think 2 will show up than 1? Some of them should have the intuition that it will be twice as often. Ask how much more often they think 3 will come up than 1.
Have them do an experiment and create a bar chart with their results. Make them do enough rolls to see the expected trend. Ask them why everyone's bar chart is not exactly the same.
Tell them that since there are six sides on the die, and one of them is marked "1", every time you roll the die, there is one chance out of six that it will land on the 1. Ask them how many chances out of six there are that the die will land on a 2. And how many chances out of six are there that the die will land on a 3. Ask them, if you roll the die 6 times, are you guaranteed to get 1 once, 2 twice and 3 three times? Since they have done the experiment they know this is not so. Ask them why they think it would or would not be guaranteed.
Propose different games to be played with this die, and ask the students if they are fair or not. For example, we roll the die once, and if it comes up 1 you win, and if it comes up 2 I win, and if it comes up 3, we roll again. Ask: Is that fair? What would be fair?
Another thing you could do (to get at equivalences and scaling) is ask the kids, if I had a 12-sided die, how should I mark the sides if I want the chances of getting 1, 2 or 3 to be the same as it was for the 6-sided die that we were just using. In other words, how could I label the die so that if you rolled that die a large number of times, your bar charts would look similar to these ones you all made with the 6-sided die. This will be tricky for them because the chances are now "out of 12" instead of "out of 6". Is 1 chance out of 6 the same as 1 chance out of 12? If not, what would be the same?
You can also try doing basic probability with a spinner. Take a game spinner on a background broken into one half and two quarters. Label the spot that covers half of the spinner area A, and the two quarters B and C. Ask the kids, if they spin the spinner a lot of times, which letter will come up most often. Ask them to compare how often they expect to get A versus B, and how often they expect to get C versus B. If you can come up with decent spinners for them to play with, you can have them do that experiment too. Ask them if they can give you a "chances out of" statement for how often B will come up. If they are stuck, try dividing A's region in half with a dotted line, so they can see 4 equal sized (and thus equally likely regions) and then ask them if they can come up with a "chances out of" statement. If they are still stuck, point out that there are 4 equally likely regions on the spinner, just like there were 6 equally likely sides on the die. So they need to figure out how many chances out of four each letter has of being chosen on the spinner.
If you want to try introducing the notion of equivalence and scaling up with the spinners, you could then divide all the segments in half again, so there are now 8 segments on the spinner. Now B looks like it has 2 chances out of 8, and before it looked like 1 chance out of 4. Are those the same thing? Can they explain why they think so?
If they are getting good at it, then you could try some "or" questions, like on the die, what are the chances of getting a 1 or a 2 on any given roll? (3 chances out of 6)
I don't see as easy a way of doing "and" questions, like what is the chance of getting a 1 the first time you roll the die and then a 2 the second time. Making the chart of all 36 combinations, especially when some look identical, will not go well in the primary grades, IMO. If you do want to get into that, get two different colored dice, normally numbered 1 through 6, and ask what is the chance of getting a 1 on the red die, and a 2 on the blue die. And then help them enumerate all 36 combinations. But honestly, I think at that level, you're better off working with slightly older kids who understand multiplication and fractions.