The image contains a mathematical question related to probability formulas. Here's the information extracted from it: --- ### Question Given: \[ P(T|D) = .98, \, P(T|D^c) = .10, \, P(D) = .05 \] Calculate: \[ P(D|T) \] ### Formula \[ P(D|T) = \frac{P(T|D) \cdot P(D)}{P(T)} \] Where: \[ P(T) = P(T|D) \cdot P(D) + P(T|D^c) \cdot P(D^c) \] ### Calculation 1. **Calculate \( P(T) \):** \[ P(T) = (.98 \times .05) + (.10 \times .95) \] 2. **Calculate \( P(D|T) \):** \[ P(D|T) = \frac{.98 \times .05}{P(T)} \] Fill in these calculations to complete the problem.

# Probability Calculation ## Given Data - \( P(T|D) = 0.98 \) - \( P(T|D^c) = 0.10 \) - \( P(D) = 0.05 \) ## Required Calculation Find \( P(D|T) \). ## Formula Using Bayes' Theorem: \[ P(D|T) = \frac{P(T|D) \cdot P(D)}{P(T)} \] Where \( P(T) \) is calculated as: \[ P(T) = P(T|D) \cdot P(D) + P(T|D^c) \cdot P(D^c) \] ## Steps for Calculation ### 1. Calculate \( P(D^c) \) \[ P(D^c) = 1 - P(D) = 1 - 0.05 = 0.95 \] ### 2. Calculate \( P(T) \) \[ P(T) = (0.98 \times 0.05) + (0.10 \times 0.95) \] \[ P(T) = 0.049 + 0.095 = 0.144 \] ### 3. Calculate \( P(D|T) \) \[ P(D|T) = \frac{0.98 \times 0.05}{0.144} \] \[ P(D|T) = \frac{0.049}{0.144} \approx 0.3403 \] ## Final Answer \[ P(D|T) \approx 0.3403 \]