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 solve this problem, we need to consider all possible combinations of rankings from the four girls and determine the probability of achieving a total score of 4 for cake A.
Let's analyze the possible rankings for each girl:
Girl I can rank the cakes as 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, or 3-2-1 (6 possibilities).
Girl II can also rank the cakes in 6 different ways.
Girl III can rank the cakes in 6 different ways as well.
Girl IV can also rank the cakes in 6 different ways.
Therefore, the total number of possible combinations of rankings is 6 * 6 * 6 * 6 = 1296.
Now, let's count the number of combinations where cake A receives a total score of 4.
If cake A receives a score of 1 from one girl, it must receive scores of 2 from the other three girls. Similarly, if cake A receives a score of 2 from one girl, it must receive scores of 1 and 3 from the other two girls, and so on.
Let's consider each case individually:
Case 1: Cake A receives a score of 1 from one girl.
There are 4 possible girls who could give cake A a score of 1. For each of these cases, the remaining 3 girls must rank cake A as 2. Therefore, the number of combinations for this case is 4.
Case 2: Cake A receives a score of 2 from one girl.
Again, there are 4 possible girls who could give cake A a score of 2. For each of these cases, one of the remaining 3 girls must rank cake A as 1 and another girl must rank it as 3. There are 3 possibilities for the girl who ranks it as 1 and 2 possibilities for the girl who ranks it as 3. Therefore, the number of combinations for this case is 4 * 3 * 2 = 24.
Case 3: Cake A receives a score of 3 from one girl.
Similarly, there are 4 possible girls who could give cake A a score of 3. For each of these cases, the remaining 3 girls must rank cake A as 2. Therefore, the number of combinations for this case is 4.
The total number of combinations where cake A receives a total score of 4 is 4 + 24 + 4 = 32.
Finally, we can calculate the probability by dividing the number of combinations where cake A receives a score of 4 by the total number of possible combinations:
Probability = 32 / 1296 = 0.0247 (approximately)
Therefore, the probability that cake A will receive a total score of 4 is approximately 0.0247, or about 2.47%.