Argument for adopting a divinely favoured way of life -named after French philosopher, mathematician, physicist and pious gambler Blaise Pascal (1623-1662) who stated it in his Pensees (§233) - but apparently stemming from Islam.
One statement of it (not Pascal's) is this. Let the utility of a policy be the gain it promises multiplied by the probability of getting it. Let the gain of a life of pleasure be some finite quantity X, certainly achievable. But if God exists he may punish our sins with infinite suffering in Hell. Let the probability that God both exists and will do this be as small as you like, but finite (above zero).
The product of an infinite loss and a finite probability is still infinite. Therefore the utility of choosing pleasure over virtue is X minus infinity; that is, is infinity. But the utility of choosing virtue is at worst (that is, if God will not reward us, and death is the end) minus X (that is, the certain loss of X units of pleasure), which is finite.
Therefore, it must be rational to choose virtue; that is one should act as if God exists because any positive probability of going to Hell and suffering infinite punishment cannot be balanced by the finite advantages gained by sin.
Unfortunately, if the argument proves anything it proves too much, since it could be reiterated for infinitely many different hypotheses about God's preferences and intentions.
P T Landsberg, 'Gambling on God', Mind (1971)