Question 1201638: In an experiment, A, B, C, and D are events with probabilities P[A∪B]=5/8, P[A]=3/8,
P[C∩D]=1/3, and P[C]=1/2. Furthermore, A and B are disjoint, while C and D are independent.
(20 points)
a) Find P[A∩B], P[B], P[A∩Bc
], P[AUBc
].
b) Are A and B independent?
c) Find P[D], P[C∩Dc
], P[Cc∩Dc
], P[C|D].
d) Are C and Dc
independent?
Answer by mathprof(1)   (Show Source):
You can put this solution on YOUR website! First let's solve for P[B] and P[D], since P[A] and P[C] are given.
P[A∪B]= P[A] + P[B] = 5/8, since A and B are disjoint.
= 3/8 + P[B] = 5/8, hence P[B] = 2/8 = 1/4
P[C∩D] = P[C]P[D] = 1/3, since C and D are independent.
= 1/2 * P[D] = 1/3, hence P[D] = 2/3
a)
P[A∩B] = 0, since A and B are disjoint.
P[B] = 1/4 (see above)
P[A∩Bc] = P[A] = 3/8, since A and B are disjoint, A is a subset of Bc.
P[A∪Bc] = P[Bc] = 1 - 1/4 = 3/4, since A and B are disjoint, A is a subset of Bc
b)
A and B are not independent, since P[A∩B]=0 does not equal P[A]P[B] = 3/8 * 1/4.
c)
P[D] = 2/3 (see above)
P[C∩Dc] = 1/6, since P[C∩D] + P[C∩Dc] = P[C], i.e., 1/3 + P[C∩Dc] = 1/2.
P[Cc∩Dc] = P[(C∪D)c]
= 1 - P[C∪D] = 1 - (P[C] + P[D] - P[C∩D])
= 1 - (1/2 + 2/3 - 1/3) = 1/6
P[C|D] = P[C] = 1/2, since C and D are independent.
d) C and Dc are independent, because P[C∩Dc] = P[C]P[Dc] = 1/2 * 1/3 = 1/6.
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