Is the flow toxic today

A market maker models each day's order flow as toxic or normal, a state it cannot see. A toxic day stays toxic the next day with probability $0.5$. A normal day stays normal with probability $0.9$. Last night's filter put toxic at $0.2$. Heavy one-sided flow appears on $80\%$ of toxic days and $20\%$ of normal days, and it appears today. What is the updated probability that today is toxic? Give a fraction in lowest terms.

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