Question

Let 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 _____.

Answer 1

Correct Answer: 24

The grid shown has the following points with their respective number of friends:

• Number of points having 0 friends = 0
• Number of points having 1 friend = 0
• Number of points having 2 friends = 4
• Number of points having 3 friends = 5 × 4 = 20
• Number of points having 4 friends = 49 − 24 = 25

The table below shows the probability (pᵢ) and the corresponding number of friends (Xᵢ) for each point:

\(\begin{array}{c|cccc} p_i & 0 & 0 & \frac{4}{49} & \frac{20}{49} & \frac{25}{49} \\ X_i & 0 & 1 & 2 & 3 & 4 \\ \end{array}\)

The expected value is given by:

\(7 \cdot E(X) = 7 \left( 0 \cdot \frac{0}{49} + 1 \cdot \frac{4}{49} + 2 \cdot \frac{20}{49} + 3 \cdot \frac{25}{49} \right)\)

Now, calculate the expected value:

\(7 \cdot E(X) = 7 \left( 0 + \frac{4}{49} + \frac{40}{49} + \frac{75}{49} \right) = 7 \cdot \frac{100 + 60 + 8}{49}\)

\(7 \cdot E(X) = \frac{168}{49} = 24\)

Thus, the expected number of friends is 24.

Answer 2

We are given that there are 49 points in total: A₁, A₂, ..., A₄₉. Two distinct points are chosen randomly from these points, and we need to determine the probability that they are friends. 

Step 1: Number of ways of selecting 2 adjacent dots in one row

The number of ways of selecting 2 adjacent dots in one row is 6. This is because in a row of 7 dots, you can select 2 adjacent dots in 6 different ways (the 1st and 2nd dot, 2nd and 3rd dot, and so on until the 6th and 7th dot).

Step 2: Number of ways of selecting 2 adjacent dots in one column

Similarly, the number of ways of selecting 2 adjacent dots in one column is also 6. This is because in a column of 7 dots, you can select 2 adjacent dots in 6 different ways (the 1st and 2nd dot, 2nd and 3rd dot, and so on until the 6th and 7th dot).

Step 3: Number of ways of selecting 2 adjacent dots from the matrix

Now, considering the entire matrix, the total number of ways of selecting 2 adjacent dots is:

\(\text{Number of ways} = 6 \times 7 + 6 \times 7 = 6 \times 7 \times 2 = 84\)

Step 4: Probability Calculation

The total number of ways of selecting 2 points from 49 points is given by the combination formula:

\(\binom{49}{2} = \frac{49 \times 48}{2} = 1176\)

The probability p that the two points are friends is the ratio of favorable outcomes (84) to total outcomes (1176):

\(p = \frac{84}{1176} = \frac{84}{49 \times 48} = \frac{2}{49}\)

Step 5: Calculation of 7p

Now, we multiply the probability by 7:

\(7p = 7 \times \frac{84 \times 2}{49 \times 48} = \frac{7 \times 84 \times 2}{49 \times 48} = \frac{1}{2} = 0.50\)

Thus, the value of 7p is 0.50.