The study by the Imperial Cancer Research Fund (ICRF) also found only 7% of people knew their chances of getting cancer were greater than one in three.
More than half thought their risk was one in 10 or even less.
Most recent records show approximately 109,410 men and 109,890 women are diagnosed with cancer each year in England and Wales.
The figures in Scotland are 13,070 men and 13,570 women.
The ICRF survey showed 19-24 year olds were the most aware of their cancer risk, with 11% accurately estimating it to be one in three.
But just 4% of 67-74 year olds were aware of the risk.
Scotland was the most aware part of the UK, with 12% getting it right - compared with just 1% in the south of England.
Sir Paul Nurse, director general of the Imperial Cancer Research Fund, said: "Many people do not appear to be aware of this harsh statistic, but we believe that it is important that people should know the true risk of cancer."
Progress in treatment
Much progress is being made in the fight against cancer.
Nine out of ten men with testicular cancer are treated successfully, and nearly 70% of children survive the most common form of childhood leukaemia.
Deaths from cervical cancer have fallen by 50% in the last 15 years.
Sir Paul said: "Real progress is being made in our knowledge of cancer, but we need to do better.
"Cancer is not one but over 200 different and very complex diseases. That's why it is such a difficult disease to understand."
It is estimated about 80% of cancers could be avoided each year by prevention alone.
Preventative measures include: taking care in the sun, giving up smoking, eating well and being screened regularly.
The ICRF is launching an advert which will appear in national newspapers and poster sites throughout the UK.
It shows three little girls, with their arms entwined - and each with a single word above her head. One reads: Lawyer; the second,Teacher; and the third, Cancer.