Rohan Chorge
Roll- 2013033
Group-5

Laws of Probability

Multiplication Law: If A1, · · · , Ak are independent

events, then

Pr(A1 ∩ A2 ∩ · · · ∩ Ak) = Pr(A1) Pr(A2)· · ·Pr(Ak).

Addition Law: If A and B are any events, then

Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)

Note: This law can be extended to more than 2 events.

Conditional Law of Probablity-Dependent Event

The conditional probability of B given A

Pr(B|A) =

Pr(A ∩ B)/Pr(A)

A and B are independent events if and only if

Pr(B|A) = Pr(B) = Pr(B|A)

'Addition Law example’

For example, assume we wish to determine the probability of drawing a king and/or a queen out of a deck of cards on only one draw. Using the addition rule for probabilities, we get the following: P(King) = 4/52, P(Queen) = 4/52, and P(King and Queen) = 0. 

Since it is impossible to draw both a king and queen on the same draw, we can conclude that the probability of drawing either a king or queen from a deck of cards is 8/52

'Multiplication  Law example’

Suppose we roll one die followed by another and want to find the probability of rolling a 4 on the first die and rolling an even number on the second die. Notice in this problem we are not dealing with the sum of both dice. We are only dealing with the probability of 4 on one die only and then, as a separate event, the probability of an even number on one die only.

P(4) = 1/6 
P(even) = 3/6

So P(4even) = (1/6)(3/6) = 3/36 = 1/12

'Conditional  Law example’

If P(A)=.5, P(B)=.4, and P(AB)=.2 (hence P(AUB)=.7 and P(A'B')=.3), P(A|B)=.2/.4=.5 and P(B|A)=.2/.5=.4.