Scoring Probabilities

Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance $x$ is exactly $(1 – x / q)$, where $q$ is a real constant greater than $50$.
During each practice run, she takes shots from distances $x = 1, x = 2, \dots, x = 50$ and, according to her records, she has precisely a $2\%$ chance to score a total of exactly $20$ points.
Find $q$ and give your answer rounded to $10$ decimal places.

To solve this problem, we can use Python code to calculate the probability of Barbara scoring exactly 20 points from distances 1 to 50, and then find the value of q that satisfies the given conditions. We can use a loop to iterate through the distances and calculate the probabilities.

Here’s the Python code to solve the problem:

“`python
import sympy as sp

q = sp.Symbol(‘q’)
total_distance = 50
target_points = 20
prob_target = 0.02

# Calculates the probability of scoring exactly n points from distance x
def calculate_probability(x, q):
return (1 – x / q)

# Calculates the total probability of scoring exactly target_points from distances 1 to total_distance
def calculate_total_probability(total_distance, target_points, q):
total_prob = 0
for x in range(1, total_distance + 1):
p_x = calculate_probability(x, q)
total_prob += sp.binomial(total_distance, x) * p_x**target_points * (1 – p_x)**(total_distance – x)
return total_prob

# Use sympy’s solver to find the value of q
sol = sp.solve(calculate_total_probability(total_distance, target_points, q) – prob_target, q)

# Round the solution to 10 decimal places and print the result
result = round(float(sol[0]), 10)
print(“The value of q is:”, result)
“`

When you run this code, it will output the value of q rounded to 10 decimal places.

Note: We are using the `sympy` library, which is a powerful symbolic mathematics library in Python, to solve the equation.

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