probability

Important Points

• $P(A\cup B)=P(A)+P(B)+P(A\cap B)$

• If a set expression feels difficult to solve, use venn diagrams to vizualize what the section of the diagram it actually is

• When intersection comes out of a set expression, the signs of the set expression changes $P(A^C\cap B^C)=P((A\cup B{)}^C)P(A^C\cup B^C)=P((A\cap B{)}^C)$

• Conditional Probability: Prob that event A happens given that B has already happened $P(A/B)=\frac{P(A\cap B)}{P(B)}$

• Multiplication Theorem: Prob of two events A and B happening simultaneously is equal to product of probability that event A happens and prob that event B happens give that A has already happened $P(A)\times P(A/B)=P(A\cap B)$ $P(A\cap B\cap C)=P(A)\times P(B/A)\times P(C/(A\cap B))$

• Independant Events $P(A/B)=P(A)$ $P(A\cap B)=P(A)\times P(B)$

• Mutually Exclusive Events: events that can’t happen together $P(A\cup B)=P(A)+P(B)$ $P(A\cap B)=0$

• Total Probability Law: It gives the probability of event A happening if there are many ways($E_1,E_2,E_3...$) to accomplish event A $P(A)=P(E_1)\times P(A/E_1)+P(E_2)\times P(A/E_2)+P(E_3)\times P(A/E_3)...$

• Bayes Theorem: It relates the prob of event A happening given that event B happened and prob of event B happening given that event A happened

  IMPORTANT

  In question of Bayes theorem, the probability of an outcome will be given, asked will be the probability that it happened a certain way

  $P(E_1/A)=\frac{P(E_1)\times P(A/E_1)}{P(A)}$