Question

What is the probability, p(A|B), where A is the roll of a “3” or a “4” on a fair die and B is the probability of tossing exactly two heads out of 4 tosses with a fair coin? It is assumed the rolls of the die and the tosses of the coins are independent. Provide your answer as a reduced fraction.

Answer 1

Consider , A fair die is rolled .

The sample space is ,

S={1,2,3,4,5,6}

Let A be the event that there is "3" or "4" on the face of die .

Therefore ,

Because Number of favourable elements to event A are TWO ,and that are "3" & "4" .

Also Consider,

A fair coins are tossed for 4 times , (This is same as tossing 4 coins simultaneosly )

The sample space is ,

S={HHHH,HHHT,HHTH,HHTT,HTHH,HTHT,HTTH,HTTT

  THHH,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT }

Let , B be the event that exactly two heads out of 4 tosses with a fair coin .

Because there are total 6 elements from all 16 elements from sample space that favouring the event B and that are

"HHTT","HTHT","HTTH","THHT","THTH","TTHH"

Now we want to find ,

Since A and B are independent ( Roll of die and coin tossed are independent ),

We can write ,

Therefore ,