In the Bhagavad Gita, Krishna says that devotional service unto Him is very very rare, so rare that even if accept Krishna as Supreme, it is impossible to understand Krishna. These are pretty tough odds to beat especially if this statement is coming directly from the Supreme Being Himself.
So let us try to "quantify" the odds in our very own idiosyncratic sense.
Our criteria for devotion is anyone who chants the Holy Names of Krishna consistently everyday because in this age of Kali, there is no other way to be His devotee. So what are the odds one will chant the Hare Krishna Mahamantra everyday in this kali yuga setting?
Odds of being born in this kali yuga
When Brahma goes to sleep, there is partial destruction of planets of this universe and when he wakes up during the day, creation takes place again. So in that sense since this day of Brahma has begin, 2 billion years have passed in earthly time. Krishna appears once every day of Brahma. It is in this 28th divya yuga in dwapara yuga of 7th manvantara Krishna has appeared and Lord Chaitanya only appears every time after the appearance of Krishna. Those reading this post happened to also be born just at the cusp of the appearance of Krishna and Chaitanya. This is important because we cannot chant the Hare Krishna Mahamantra without Lord Chaitanya.
So the odds to be born in this kali yuga just after Krishna and Chaitanya is 1 in 2 billion.
Odds of being born as a human being
In the Padma Purana, it is said that there are 8.4 million species of life (84 lacs). For the sake of this calculation, we will assume we take birth once in each species (this is unlikely because typically we take multiple births within the same species).
So the odds to be born as human is 1 in 8.4 million.
Odds of meeting a devotee
Currently, there are estimated 7 billion humans in this world. Of them say 3.5 billion are between 25 to 64 (active adults). If anything we would most likely come in touch with a devotee within that age range. Of course Prabhupada was beyond that age when he initiated this whole movement but of course Prabhupada is special. Of the 3.5 billion, I estimate there would be 1 million devotees in that age range (this is very liberal the odds actually will be much higher)
So the odds of meeting a devotee is 1 in 1 million
Combining all, the odds of being born in this kali yuga and meeting a devotee is 1 in 2.01 billion. Assuming you meet a devotee and take up chanting seriously, you are looking at an odd akin to winning the powerball lottery at least 11 times in one life. This is equivalent to a prize money of 440 million dollars.
This is an illustration to show that on our own, one cannot serve and love God because the odds are stacked against us. To the degree we are free from false-prestige and pride, and with utter humility and tolerance take shelter of Krishna, to that degree Krishna will nourish the seed of bhakti within our heart.