If P\left(A\right)=0.4, P\left(B\right)=0.5, and P\left(A\cap B\

Jessica Scott

Jessica Scott

Answered question

2021-11-20

If P(A)=0.4, P(B)=0.5, and P(AB)=0.3, find a. P(AB), b. P(AB), and c. P(AB).

Answer & Explanation

James Kilian

James Kilian

Beginner2021-11-21Added 20 answers

Explanation 1
Step 1
PROBABILITY RULES
Complement rule
P(A)=P(¬A)=1P(A)
General addition rule for any two events:
P(AB)=P(A)+P(B)P(AB)
Step 2
SOLUTION
P(A)=0.4
P(B)=0.5
P(AB)=0.3
(a) Use the general addition rule for any two events:
P(AB)=P(A)+P(B)P(AB)
=0.4+0.50.3
=0.6
Explanation 2
Step 1
Probability rules, used n the exercise:
Complement Rule:
(A)=P(¬A)=1P(A)
Addition Rule of general events:
(AB)=P(A)+P(B)P(AB)
Rule of Complementary events B and B':
P(A)=P(AB)+P(AB)
De Morgan's Rules
P((AB))=P(AB)
P((AB))=P(AB)
Step 2
Solution:
P(A)=0.4
P(B)=0.5
P(AB)=0.3
(a)
(AdditionRule)P(AB)=P(A)P(AB)
=0.4+0.50.3
=0.6

Crom1970

Crom1970

Beginner2021-11-22Added 6 answers

Explanation 1
Step 3
(b) The event of A occurring but not B is the event A without all outcomes that occur for A and B have in common.
P(AB)=P(A)P(AB)=0.40.3=0.1
Explanation 2
(b)
(Rule of Complementary events)P(AB)=P(A)P(AB)
0.40.3
=0.1

Don Sumner

Don Sumner

Skilled2021-11-23Added 184 answers

Explanation 1
Step 4
(c) The event of A not occurring or B not occurring is the event that A and B cannot occur together.
Use the complement rule: 
P(AB)=P((AB))=1P(AB)=10.3=0.7
Explanation 2
(c)
(DeMorgansRule(2))P(AB)=P((AB))(ComplementRule)=1P(AB)
=10.3
=0.7

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