To sum up: children have same expectations of odds, probabilities, and equity as adults so long as the problem is stated clearly. For best interest capture, make it even odds (along the probabilities of blackjack) as influenced by a player-controlled simple skill minigame per test. However, the problems with Piaget's study do suggest maximal elimination of the "character-building meta-game" as much as possible.
Looking at "Children's Understanding Of Randomness: Report Of A Survey
Of 160 Children Aged 7-11 Years" (Green 1986):
The high facilities for the random and regular patterns indicate that young
children do have a sound conceptual awareness of randomness. The lower
facility for the semi-random pattern shows that children's lack of understanding
of a problem may produce responses which mislead the investigator.
To translate: there is a strong intuitive perception of the fact of randomness in primary school aged children, but this fact may be occluded by poor design. Therefore, in spite of Piaget's (1975) claims to the contrary, this question in fact, has merit.
Looking at Engel and Sedlmeier (2005):
Various studies indicate
that children have valid intuitions about probability. ... All these results indicate an
increasing and progressive improvement in statistical
judgment.
This paper also disproves Piaget's observations on raindrop distribution.
Therefore, we can presume that a child's hedonic response to success may be directly proportionate to an adult's, so long as the illusion of skill is preserved. In this instance, there should be multiple routes to success to provide for poor and strong impulse control, with equivalent odds weighted for either long term success or (smaller) short term gains. It's also critical to have a minigame in the die rolling, as small skill-based competencies are highly attractive.
(As an aside, may I note that the literature in this area is extremely depressing.)
Looking at this thesis, there's a strong preference for blackjack in "common casino activities." The standard perception of blackjack is a 50% odds per hand (less the house's cut, with strategy possible in doubling and whatnot.) Therefore, have your skill minigame work out to perfectly fair play (50/50) influenced strongly by player choice while rolling the dice. Optimal minigame play should work out to better than even odds (it doesn't have to be significant, call it a 10-20% variation (test this variation strongly in your desired groups) for most optimal to least optimal play).
This design pattern provides for frequent tests of personal skill (which act as useful distractions as attention span is governed by development). One common theme in the papers I read is that personal competence/skill was an important feeling in compulsive gamblers. Combine that with the attraction of new fruit machines to young children by incorporating skill elements, and the ideal of blackjack which involves a skill-decision at the crucial point.
The only solid analysis on risk-skill desirability I could find was a model suggesting that "motive, expectancy, and incentive" are factors which suggest "that performance level should be greatest when there is greatest uncertainty about outcome." Therefore, people with strong motive to achieve should prefer immediate risk whereas those with strong motive to avoid failure will prefer easy tasks or extremely difficult and risky tasks (Atkinson 1957).
As most of the literature is on how to limit gambling in young adults/children (see depressing aside), which is why I had to prove the same odds perception capabilities as adults, then look for risk preference in adults. A reverse citation search of Atkinson's impact on game design is left to the reader.
Someone actually in young child development will do a better keyword search, but there's too much anti-gambling noise in nominal lit searches. Most skill/luck differentials were related to compulsive/non-compulsive gamblers interacting with the same gambling task. I merely framed my suggestion to correspond with the gambling identities proposed by children with compulsive gambling problems. (As it's better to actually provide a real consequence of skill, then allow it to be a justification of a problem behaviour.)
With rules articulated, there is an expressed preference for equity in young children, and a desirability of some risk (as strongly learned/inherited) from parents. However, there was an expressed interest in greater payoffs for competence and motivation, with similar patterns to adult preference for equity. Again, a strong theme in the literature is make expectations clear, children (especially very young children) don't comply with (due to ignorance or other reasons) implicit social sharing rules. (Nelson and Dweck, 1977)
As an aside, fast turn resolution and distraction management are critical to your game's success here.
