One way to look at current reliability metrics is ETOPS. Twin engine airliners are licensed to fly remote routes by meeting ETOPS standards. This dictate how many minutes away from a diverting airport they can fly, and it's based on the percentage likelihood they'll have an in-flight engine shut-down in 1,000 hours of flight. They need to achieve:
5% per 1000 engine hours for EDTO 120 minutes.
2% per 1000 engine hours for EDTO 180 minutes.
1% per 1000 engine hours for EDTO beyond 180 minutes.
Now most airliners don't meet these standards, and they aren't required for four engined planes (which I believe have different standards), but for the sake of this analysis lets assume 747 engines are at 5%.
A rough range of pricing for jet engines for commercial airliners is \$12M to \$35M, a 747 is obviously fitted with engines from the high end of the scale, so lets assume \$120M for a set of four. Assuming a 20 year life span, that's $6M a year in depreciation.
Now our new engine company (SuperCheapTurbines!, inc) shows up with our super duper low cost engines that provide the same performance as the \$30M engines for only \$1.5M each. The only trade off is they average a failure once every 1,000 hours instead of 20,000 hours but the airline will save $5.7M a year in depreciation. Note: I'm using 20x cheaper and a 20x higher fail rate for a more realistic comparison.
So how many do we sell, and do we decide to go public immediately or sell out to Boeing for billions? The answer is none, no, and no. Every airline declines our engines because paradoxically they say they are too costly!
Think about how that 747 is being used. Assume it's averaging 12 hours of flight time per day, and typically for a single long distance flight with an average of 300 passengers at \$500 each in revenue. So it's generating $150,000 per flight/day in revenue, or over \$50M a year. When it flew with the expensive engines, their failure rate meant it had an engine shutdown in flight roughly once every 400 days.
But with our engines fail about 10% of the time every 100 hours, and given there are four of them, the 747 would have a 35% chance of a shutdown engine every 100 hours. Here's the problem, while often airlines have the discretion to continue a flight with one engine out on a four engine airliner given the massive reliability of current engines, it's still a dangerous situation. If another engine goes out it will become a full fledged emergency, but for 5% engines a second engine failure would only be expected once every 1,000 - 2,000 single engine out occurrences.
But given how unreliable our engines are, there is virtually no chance regulators and airlines would permit flights to continue with only 3 engine operation, they would flag it as emergency requiring immediate abort of the flight. If one of our engines fails, there is roughly a 1 in 60 chance a second engine will fail on the same flight. So every engine out for a 747 equipped with SuperCheapTurbines! is likely an aborted flight.
And we'll have a lot of aborted flights. SuperCheapTurbines! equipped 747s will average an engine out once every 300 hours, or once every 25 days, costing over \$2M in lost revenue a year, and millions more in replacement engines. And that's just the start, every abort creates a huge amount of additional costs:
Repairing the engine.
Carrying more expensive backup 747s.
Lower fares because of the airlines terrible reliability.
And occasionally an engine out situation cascades into a triple engine out or other emergency, losing a plane with crew and passengers and costing the airline hundreds of millions of dollars in settlements and in replacing the plane. And further in reputation.
Essentially this is why air travel has advanced so much from the propeller days. Back then to fly long distances safely designers added more engines, but the more engines they added the more likely an engine out situation became, or worse, an engine fire. Powerful, reliable jet engines have allowed airlines to switch almost entirely to twin jets that mathematically have far fewer engine out situations than three and four jet engine airliners, which means they can fly long distances safely even over water.
Note: I skipped over some things here, including normal maintenance costs which I assume are similar, and the higher interest costs for financing more expensive engines because I don't think they change much in the math.