2. multiplication theorem on probability

Multiplication theorem on probability

we can write

P(E ∩ F) = P(F) . P(E|F)    ………….... (1)

P(E ∩ F) = P(E). P(F|E)        ............... (2)          

The above result is known as the multiplication rule of probability.

Multiplication rule of probability for more than two events If E, F and G are three events of sample space, we have P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)

Example:

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

Solution:

Let K denote the event that the card drawn is king and A be the event that the card drawn is an ace. Clearly, we have to find P (KKA) Now P(K) = 4 /52 Also, P (K|K) is the probability of second king with the condition that one king has already been drawn. Now there are three kings in (52 − 1) = 51 cards. Therefore P(K|K) = 3 /51 Lastly, P(A|KK) is the probability of third drawn card to be an ace, with the condition that two kings have already been drawn. Now there are four aces in left 50 cards.

Therefore P(A|KK) = 4 /50 By multiplication law of probability,

we have P(KKA) = P(K) P(K|K) P(A|KK) = (4/52) X (3/ 52) X (51/ 50) = 2/5525