There are 4 students in a class. A teacher wants them to each secretly choose a partner for a group project. If everyone independently chooses a partner randomly, the probability that everyone chooses a partner who chose him/her is P.
To find the probability that everyone chooses a partner who chose him/her, we need to consider the possible outcomes and divide them by the total number of possible outcomes.
Let's break it down step by step:
Step 1: Determine the total number of possible outcomes.
In this case, each student has 3 other students to choose from as a partner. So, for the first student, they have 3 choices, for the second student, also 3 choices, for the third student, again 3 choices, and for the last student, the remaining 1 choice. Therefore, the total number of possible outcomes is 3 x 3 x 3 x 1 = 27.
Step 2: Determine the number of favorable outcomes.
For everyone to choose a partner who chose him/her, the only possible outcome is when each student chooses the student sitting directly across from them. Therefore, there is only 1 favorable outcome.
Step 3: Calculate the probability.
The probability (P) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
P = Number of favorable outcomes / Total number of possible outcomes
P = 1/27
So, the probability that everyone chooses a partner who chose him/her is 1/27.