Law of total probability

Law of total probability


P(A and B)=P(A)\cdotP(B|A) or equivalently, P(A and B)=P(B)\cdotP(A|B)

The Law of Total Probability:
P(A)=P(A and B)+P(A and ~B)=P(B)P(A|B)+P(~B)P(A|~B)

Or in full generality, if all of B1,B2,...BnB_1, B_2,...B_n include the entire sample space S, and are all pairwise mutually exclusive then:

P(A)=P(AP(A)=P(A and B1)+P(AB_1)+P(A and B2)++P(AB_2)+ \cdots +P(A and Bn)B_n)
=P(B1)P(AB1)+P(B2)P(AB2)++P(Bn)P(ABn)=P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots + P(B_n)P(A|B_n)
  • Introduction
    Illustrating the Law of Total Probability for two conditional events

    The Law of Total Probability for more than two events

  • 1.
    2 disjoint mutually exclusive events
    Back Country skiers can be divided into two classes, those with avalanche training and those who have no avalanche training. An individual with avalanche training has a probability of 0.05 of getting in an avalanche, while an individual without avalanche training has a probability of 0.20 of being in an avalanche. If 75% of backcountry skiers have avalanche training and 25% of back country skiers do not, then what is the probability that a randomly selected backcountry skier will be in an avalanche?

  • 2.
    3 or more disjoint mutually exclusive events
    I have 3 bags that each contains 5 marbles.

    Bag 1:

    Bag 2:

    Bag 3:

    2 Green

    4 Green

    5 Green

    3 Red

    1 Red

    0 Red

    I choose one bag at random and draw a marble. What is the probability that I draw a green marble?

  • 3.
    Thomas is frequently late to work. If it is sunny he will be late with probability 0.15; if it rains he is late with probability 0.05. And if it snows he is late with probability 0.5 (if he shows up at all). The meteorologist has predicted there is a 50% it will be sunny tomorrow, a 35% chance it will rain and a 15% chance it will snow. What is the probability that Thomas will be late tomorrow?

    Make jokes about taking sick days + make pictures for the sun, rain and snow.