One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is

Solution

Maths sir · 3 steps

Label every face separately so all 36 face pairs are equally likely, then count faces.

1. Equally likely outcomes. Each die has 6 faces, so there are 6 × 6 = 36 equally likely face pairs. The number pair (1, 3) can happen in 2 × 2 = 4 ways, because die 1 has two faces marked 1 and die 2 has two faces marked 3.
2. Faces per number. Die 1 has 2, 2, 1, 1 faces showing 1, 2, 3, 4. Die 2 has 1, 2, 2, 1 faces showing 1, 2, 3, 4. Multiply the two face counts for each number pair.

   Sum	Number pairs (die 1, die 2) and ways	Total ways
   4	(1, 3): 2 × 2 = 4; (2, 2): 2 × 2 = 4; (3, 1): 1 × 1 = 1	9
   5	(1, 4): 2 × 1 = 2; (2, 3): 2 × 2 = 4; (3, 2): 1 × 2 = 2; (4, 1): 1 × 1 = 1	9

   P(sum 4 or 5) = (9 + 9)/36 = 18/36 = 1/2

3. So. The two sums cannot happen together, so you add their counts. The probability is 1/2, which is option (d).

The idea to keep. When numbers repeat on a die, count faces, not numbers. Each number pair occurs (faces with that number on die 1) times (faces with that number on die 2) ways, out of 36.

Answer from the NTA final answer key. Solution by PrepWiser.