In the modern day and age, it is very difficult to put a numerical value to this type of question, simply because the chance of something happening is so incredibly remote. This is not a clear-cut equation and is dependant on several factors.
A few thoughts:
The impact of loosing one engine will be less significant on a four-engined aircraft than a two-engined aircraft. You only loose 25% of thrust as compared to 50%. However, two engined jets are built with greater safety margin and are able to handle an engine failure at any phase in the flight. Four engined jets are certified to be able to fly on two engines on one side of the wing.
In theory, you are twice as likely to suffer an engine failure with four engines, since you have twice as many engines. While the lost thrust would be less significant, very rarely damage can propagate to cause damage to other parts (See QF32 and ELY1862). In general, it is easier, cheaper and faster to maintain two engines than four. Should a complete engine swap be necessary for instance, you only need to do it twice instead of four times. Every extra step is an step where something could theoretically could go wrong.
The GE90 (Boeing 777) is certified to handle 'large' birds (4kg) as compared to the CFM56 (Airbus A340) only being able to handle 'medium' birds. Hence, the Boeing 777 engines are likely to be a bit more robust.
From a human perspective, pilot workload will be lower with two engines rather than four. Should something happen in a time-critical emergency, there will be more systems you need to keep an overview of. See CAL006.
Furthermore, not every engine-related incident is spared simply cause you have four engines. BA Flight 9 lost all four engines due to volcanic ash. On BA Flight 38, the engines failed in unison due to ice in the heat exchanger on the engines that were operating in identical conditions.
Despite ETOPS being around for several decades and applying to thousands of aircraft I have not heard about a single accident involving commercial jets operating in that domain.
From a strictly "engine cut out" point of view in todays day and age you are not getting your self very much. There is a great article here that talks quite a bit about this point from a GA stand point. There is of course the simple fact that for some planes you need more thrust to simply fly. There has also been lots of research done on if servicing an engine frequently makes it safer or actually adds to the problem. Again from a GA (mainly piston) standpoint there is a case to be made.
The more engines you have, the better off you are if one engine fails; however, the more likely it is that one will fail.
Engines, like many things, are rated in terms of "mean time between failures" or MTBF. For aircraft engines, MTBF is relatively high (tens of thousands of hours when properly maintained), but they do still fail in relatively unassuming situations.
It's simple probability that, given a nonzero chance for a given engine to fail, the chance that at least one engine will fail is the combined chance for each particular engine to fail. So if the engine type used on a 747 has a 1/1000 chance to fail on a particular flight (the actual chance is much lower, in the tens of millionths), there is a 4/1000 = 1/250 chance of at least one engine on the plane failing during that flight. If the two engines on a 737 each had the same chance for failure, the chance of one engine failing on the flight are half those of one engine failing on the 747, because there are half the number of engines that could fail. In MTBF terms, we can say the mean time between single engine failures on a 747 is half that of a 737.
So, we see that increasing the number of engines, all other things being equal, actually increases the chances of something going wrong. However, the pilot of a twin-engined craft, though he may be twice as likely to experience an engine failure compared to the pilot of a single-engine craft, is going to have a much better day when it happens than the single engine pilot will.
The big question that really has to be asked is, how many engines can the plane fly on, and how many more than that number does it have? Jets like the DC-10 and 727 were marketed for their relatively high fault-tolerance; they could maintain level flight at low altitude on only one engine, having lost the other two, and on the 727 it didn't really matter which one was still running as they were all so close to the centerline. The 747 requires at least two of its four engines to maintain level flight (see British Airways Flight 9 in 1982, during which all four engines quit mid-flight due to ingestion of volcanic ash; restarting one engine slowed the rate of descent but it took two to level off).
Now, primarily for fuel efficiency reasons, passenger jets are designed with as few engines as they need; engines are most efficient at a specific point in their power curve, and you design the aircraft to cruise efficiently at that power output of its engines. This is why the 737 (and newer jets like the 777 and 787) doesn't have four engines; it doesn't need them, because the 737 is a fifth the weight and the engines it does have are more efficient than the 747's.
If an engine fails, the fraction of loss of thrust is smaller for an aircraft with four engines than for one with two. If other things happen, for example loss of rudder, yaw control is easier with three remaining engines than with one. So in terms of %-loss of thrust and what this entails, a 4-engine plane is safer than a 2-engine plane.
ICAO regulations allow two-engine aircrafts to fly over oceanic routes (they must have a Ram Air Turbine - RAT) so such a plane - certified as ETOPS, Extended-range Twin-engine Operational Performance Standard - must be safe. Anyway I can agree with other answers stating that if you have to loose an engine it is better to have 3 other.