Law of total probability

Recall:
P(A and B)=P(A)

\cdot

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

\cdot

P(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

B_1, B_2,...B_n

include the entire sample space S, and are all pairwise mutually exclusive then:

P(A)=P(A

and

B_1)+P(A

and

B_2)+ \cdots +P(A

and

B_n)

=P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots + P(B_n)P(A|B_n)

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?

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

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.