Ok, I think I understand what you mean by "precarious". It sounds as if the poor have a stable level of consumption (or at least a stable minimum level of consumption), but the specific manner in which that consumption is paid for can vary significantly - one week benefits, the next week employment. Also a high risk of crime, due to inadequate police protection. Is that a fair assessment of what you mean by "precarious"?
Now for some reason, the weekly variability of income and the risk of crime causes the poor to value an infinitesimal chance at large sums of money a more strongly than immediate and certain consumption. Assuming the poor are being rational here (rather than simply innumerate), this suggests that the poor assign a very low value to additional consumption, or perhaps a very high value to purely psychological benefits.
I have a suggestion for improving the lot of the poor in the UK: your government should sell more lottery tickets and use the profits to pay for additional policing. Perhaps you should also try to encourage microfinance - in the US, nonviolent microfinance outfits offering "payday loans" have pretty much pushed loansharks out of the market.
The government were given the chance to have the lottery run for free by a group headed by Richard Branson - they chose instead to maintain the current incumbent who take 5% of gross takings ... that could have paid off some tax bills ...