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7146333 (1)

# 7146333 (1) - 1 4 4 P(R|T)= 7 3 P(W|T)= 7 6 P(T)= Let P(R|...

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Unformatted text preview: 1 4 4 P(R|T)= 7 3 P(W|T)= 7 6. P(T)= Let P(R| P(R P( ¯ R ¯ T )=x and P(R)=y T)=P(R|T)*P(T)= 1 7 T)=P(W|T)*P(T)= 3 28 Getting the last two parts os a little trickier. We will do so by forming equations. ¯ ¯ T )=P(T |R)*P(R)= 3 y Using the conditional distribution formula. 4 ¯ T )=1-P(R T) ¯ P(R By De Morgan's Law ¯ T is the set of roses that are neither red nor have thorns. This is exactly ¯ R P(R equal to the set of white roses without thorns (W\W T). Also, W\W T is disjoint from W T (qed). So we can use P(W\W T)+P(W T) = P(W). Lastly, P(R)=y, so P(W)=1-y. P( ¯¯ R T )=P(W\W T) = P(W)-P(W T)=1-y-(P(W|T)P(T))=1-y- 3 y28 P(R T)=P(R)+P(T)-P(R T) =y+ 1 4 − 1 7 =y+ Substituting back 3 3 1 − y − 28 = y + 28 ⇒ 2y = 1 − 11 ∴ P (R) = 14 ¯ ¯ P(R T )=P(T |R)P(R)= 11 P( 3 ¯¯ R T )=1-y- 28 = 1 − 70 11 3 14 − 28 3 14 = = 11 14 28−25 28 1 = 3 28 3 28 31 7 ∗ 4 =1- ...
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