Let $x$ be the probability of a vegan dying from colon cancer. According to the given results:

• $P(\text{Vegan dying}) = x$
• $P(\text{Vegetarian dying}) = 2x$
• $P(\text{Omnivore dying}) = 4x$ (since vegetarians are half as likely as omnivores, omnivores are twice as likely as vegetarians)

Initial population probabilities:

• $P(\text{Omnivore}) = 0.75$
• $P(\text{Vegetarian}) = 0.15$
• $P(\text{Vegan}) = 0.10$

Using Bayes' Theorem to find the probability that the deceased was an omnivore: $P(\text{Omnivore | Died}) = \frac{P(\text{Died | Omnivore}) \times P(\text{Omnivore})}{P(\text{Died})}$

Total Probability of Dying: $P(\text{Died}) = (4x \times 0.75) + (2x \times 0.15) + (x \times 0.10) = 3x + 0.3x + 0.1x = 3.4x$

$P(\text{Omnivore | Died}) = \frac{3x}{3.4x} = \frac{3}{3.4} \approx 0.88235 \text{ or } 88.23\%$