You are handed 5 pieces of silverware for the formal setting shown. If you guess their placement at random, what is the probability that the knife and spoon are placed correctly?
To find the probability that the knife and spoon are placed correctly, we first need to determine the total number of possible arrangements and the number of favorable arrangements.
In a formal table setting, the knife is usually placed to the right of the plate and the spoon to the right of the knife.
The total number of possible arrangements can be calculated using the formula for permutations:
Total arrangements = 5!
= 5 x 4 x 3 x 2 x 1
= 120
Now, let's consider the favorable arrangements where the knife and spoon are placed correctly. Since the knife is placed to the right of the plate, there are 4 remaining places to put the knife and 3 remaining places to put the spoon.
The number of favorable arrangements can be calculated using the formula for permutations as well:
Favorable arrangements = 4! x 3!
= 4 x 3 x 2 x 1 x 3 x 2 x 1
= 144
Finally, we can calculate the probability of the knife and spoon being placed correctly by dividing the number of favorable arrangements by the total number of possible arrangements:
Probability = Favorable arrangements / Total arrangements
= 144 / 120
= 6/5
= 1.2
Therefore, the probability that the knife and spoon are placed correctly is 1.2.
To solve this problem, let's start by understanding the context. In a formal table setting, there are typically 5 pieces of silverware: a dinner fork, a salad fork, a knife, a soup spoon, and a dessert spoon.
Next, let's identify the desired outcome for our question. We want to calculate the probability that the knife and spoon are placed correctly when guessing randomly, meaning the knife is placed to the right of the dinner plate and the spoon is placed to the right of the knife.
To calculate the probability, we need to determine the total number of possible outcomes and the number of favorable outcomes.
1. Total number of possible outcomes: Since we are arranging 5 pieces of silverware randomly, the total number of possible outcomes is given by the factorial of 5, denoted as 5!.
5! = 5 × 4 × 3 × 2 × 1 = 120
2. Number of favorable outcomes: To determine the number of favorable outcomes, we need to consider the position of the knife and the spoon.
There are two possible positions for the knife: on the left of the spoon or on the right of the spoon. If the knife is on the left of the spoon, it means that the spoon is placed incorrectly. If the knife is on the right of the spoon, it means that both pieces are placed correctly.
Therefore, there is only 1 favorable outcome out of the 2 possible positions for the knife and spoon.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 1 / 120
= 1/120
Hence, the probability that the knife and spoon are placed correctly when guessing randomly is 1/120.