Probability
PYP_JEE_ADV_2023_P2
Grade None
Question:
**PARAGRAPH \"II\"**\nConsider the $6 \times 6$ square in the figure. *[Image: A 6x6 square grid with 49 intersection points (dots) arranged uniformly]* Let $A_1, A_2, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.\n\nLet $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i = 0, 1, 2, 3, 4$. Let $X$ be a random variable such that for $i = 0, 1, 2, 3, 4$, the probability $P(X = i) = p_i$. Then the value of $7E(X)$ is
Step-by-Step Solution
Key Concept: Calculating the expected value of a discrete random variable based on geometric properties of a grid.
**Step 1: Count the total number of points**
The grid consists of $6 \times 6$ smaller squares, which means there are $7$ points in each row and $7$ points in each column. The total number of points is $N = 7 \times 7 = 49$.
**Step 2: Categorize the points based on number of friends**
Points are \"friends\" if they are adjacent horizontally or vertically.\n- **Corners**: There are 4 corner points. Each has 2 friends (one in row, one in column). So, 4 points with 2 friends.\n- **Edges (non-corner)**: There are 4 edges, each with 5 non-corner points. Total = $4 \times 5 = 20$ points. Each has 3 friends (two along the edge, one inwards).\n- **Interior**: The remaining points are interior. There are $5 \times 5 = 25$ interior points. Each has 4 friends (left, right, up, down).
**Step 3: Calculate probabilities p_i**
Since each point has an equal chance of being chosen, the probabilities are:\n$p_0 = P(X=0) = 0$\n$p_1 = P(X=1) = 0$\n$p_2 = P(X=2) = 4/49$\n$p_3 = P(X=3) = 20/49$\n$p_4 = P(X=4) = 25/49$
**Step 4: Calculate the expected value**
The expected value is $E(X) = \sum_{i=0}^4 i \cdot p_i = 2\left(\frac{4}{49}\right) + 3\left(\frac{20}{49}\right) + 4\left(\frac{25}{49}\right)$.\n$E(X) = \frac{8 + 60 + 100}{49} = \frac{168}{49} = \frac{24}{7}$.
**Step 5: Calculate the final required value**
We need to find $7E(X)$.\n$7E(X) = 7 \times \frac{24}{7} = 24$.
Correct Answer: 24