Comparing Dartboard Probabilities Using Area Models | Geometry

Understanding Probability with Area Models

• 🎯 The probability of a dart landing in a specific area is calculated as the area of the shaded region divided by the total area of the dartboard.
• 📏 For a 10 cm by 10 cm square dartboard, the total area is 100 square centimeters.
• ⚠️ Simply counting shaded sections is insufficient if the sections do not have equal areas.

Calculating Gray Area on Left Dartboard

• 💡 An inner square inscribed within a larger square, with corners at the midpoints, has half the area of the outer square.
• 🧩 The gray area in the left dartboard is calculated by taking the area of the inner square (50 sq cm) and subtracting the area of the white square within it (25 sq cm), resulting in 25 square centimeters of gray area.
• 📊 The probability of landing in the gray area on the left dartboard is 25 sq cm / 100 sq cm, which equals 1/4.

Calculating Gray Area on Right Dartboard

• 📐 The right dartboard features two quarter circles removed from the total area.
• 🚀 The radius (x) of these quarter circles can be found using the Pythagorean theorem: 2x² = 10², leading to x² = 50, so x = √50.
• ⭕ The combined area of the two quarter circles (a half-circle) is (1/2) * π * r² = (1/2) * π * 50 = 25π square centimeters.
• ⚖️ The gray area on the right dartboard is the total area minus the area of the two quarter circles: 100 - 25π square centimeters.

Comparing Probabilities

• 📈 Since π is greater than 3, 25π is greater than 75. Therefore, 100 - 25π is less than 25 square centimeters.
• 🎯 The left dartboard has a gray area of 25 sq cm, while the right dartboard has a gray area less than 25 sq cm.
• ✅ Consequently, there is a higher probability of landing in the gray area on the left dartboard.