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Jung’s own explanation of the concept of synchronicity is as follows:

" As its etymology shows, this term has something to do with time, or to be more accurate, with a kind of simultaneity. Instead of simultaneity we could use the concept of a meaningful coincidence of two or more events, where something other than the probability of chance is involved." (Synchronicity, An Acausal Connecting Principle, by C G Jung, from the article "On Synchronicity", Appendix, pg. 104)

What did Jung state about the chance occurrence of coincidences?

Jung had been studying synchronicity since the middle twenties, when he was investigating the phenomena of the collective unconscious and kept on coming across connections which he simply could not explain: "as chance groupings or ‘runs.’ What I found were ‘coincidences’ which were connected so meaningfully that their ‘chance’ concurrence would represent a degree of improbability that would have to be expressed by an astronomical figure." Synchronicity, An Acausal Connecting Principle, by C G Jung, pg. 21.

Did an insect fulfill a dream?

One of the synchronicity episodes that really intrigued Carl Jung occurred with a patient recounting her dream of a golden scarab beetle. A young woman that Jung had been treating told him of a dream in which she had been given a golden scarab. He was sitting with his back to a closed window while she told him the dream.

Suddenly he heard a gentle tapping and turned to see a flying insect knocking against the window-pane from outside. Jung opened the window and caught the creature in the air as it flew in. The insect closely resembled a golden scarab. In actuality it was a scarabaeid beetle, the common rose-chafer (Cetonia aurata).

Normally, such insects would not come into a dark room, but at the same moment the patient mentioned the beetle she had been given in her dream, this beetle appeared at Jung’s window. He was quite astonished at the coincidental timing and admitted that nothing like it ever happened to me before or since. Synchronicity, An Acausal Connecting Principle, by C G Jung ,Pg. 22.

Do dreams come true?

Another fascinating synchronicity story related by Carl Jung described a dream of a student friend that came true. His friend’s father had promised him a trip to Spain if he passed his final examination. His friend was so excited about his upcoming trip that he began to dream about being in Spain. He saw himself walking through a Spanish city in a street that led to a square with a Gothic cathedral. He then turned a corner, into another street where he saw an elegant carriage drawn by two cream-coloured horses. When Jung’s friend woke up, he told a group of companions at dinner about the dream. Shortly afterwards, he successfully passed his examinations, and went to Spain. While he stood on one of the streets, he recognized the city of his dream. He was astonished to find the square, the cathedral, and the carriage with the two cream-coloured horses. Everything he saw in real life exactly corresponded to the dream images. Synchronicity, An Acausal Connecting Principle, by C G Jung, pg. 106.

http://occult-advances.org/nc-spi-Synchronicity-strange-coincidence-conspiracy.shtml