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You are told that:

  • $P(P) = 0.15$
  • $P(T|P) = 0.91$
  • $P(T|¬P) = 0.04$
  • $P$ and $T$ are not independent.

Whats $P(T)$ ?

I've been really struggling with this one. I've tried substituting a bunch of stuff into the multiplication rule $$P(A \cap B) = P(A) \cdot P(B|A)$$ and the addition rule $$P(A \cup B) = P(A) + P(B) − P(A \cap B).$$ I've also tried drawing a conditional probability tree to help me see what's going on, but I just don't seem to be getting anywhere.

Any level of help would be appreciated :)

the_candyman
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Max Randle
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1 Answers1

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$P(T)=P(T\cap P)+P(T\cap P^{c})=P(T|P)P(P)+P(T|P^{c}) P(P^{c})=P(T|P)P(P)+P(T|P^{c}) [1-P(P)]$.

[I have typed $A^{c}$ for the complement of $A$].

Kavi Rama Murthy
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  • Yes! This makes a lot of sense actually. The intuitive piece of understanding that was missing is the first line in your response. – Max Randle Mar 29 '21 at 19:55