Four girls (say I, II, III and IV) are to rank 3 different types of cake (say A, B and C) in a blindfold test. Each girl ranks the three cakes as 1 (for the cake she likes the best), 2 and 3, and then the assigned ranks of each type of cake are summed. Assume that each girl is assigning her rankings at random, what is the probability that the cake A will receive a total score of 4.

To calculate the probability that cake A will receive a total score of 4, we can consider all the possible rankings that the four girls can assign to the three types of cake (A, B, and C).

Let's break it down step by step:

Step 1: Determine the total number of possible rankings.
Since each girl can rank the cakes from 1 to 3, there are 3 options for each girl. Hence, the total number of possible rankings is calculated as 3^4 = 81.

Step 2: Find the number of favorable outcomes.
To get a total score of 4 for cake A, exactly two girls must rank it as 1, while the remaining two girls rank it as 2 or 3.

Case 1: Exactly two girls rank cake A as 1 and the other two rank it as 2.
You can choose any two girls out of the four in a total of C(4, 2) ways, which is equal to 6. The other two girls will then rank cake A as 2 (1 choice) and rank cake B or cake C as 1 (2 choices for each girl). Hence, the number of favorable outcomes for this case is 6 * 1 * 2^2 = 24.

Case 2: Exactly two girls rank cake A as 1 and the other two rank it as 3.
Using the same reasoning, the number of favorable outcomes for this case is 6 * 1 * 1^2 = 6.

So, the total number of favorable outcomes is 24 + 6 = 30.

Step 3: Calculate the probability.
The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes. Hence, the probability that cake A will receive a total score of 4 is 30/81.

Simplifying the fraction, we get the final answer:
Probability = 10/27 (approximately 0.3704)

Therefore, the probability that cake A will receive a total score of 4 is approximately 0.3704 or 37.04%.