Question: A scientist rolls a fair 10-sided die 5 times. What is the probability of rolling exactly one 7 and no 10s?

Tue, 05 May 2026 09:20:26 +0000

**1. Intro (Discover Hook)**
When a scientist rolls a fair 10-sided die five times, the question arises: What’s the chance exactly one roll shows a 7 and no roll comes up a 10? This isn’t just a dice puzzle—tracking probabilities connects directly to everyday decisions in risk analysis, game design, and statistical modeling. As data literacy and probabilistic thinking grow in the US, intriguing questions like this reflect a broader public fascination with how chance shapes outcomes. The blend of chance, logic, and real-world application makes this a naturally engaging topic for mobile users seeking clarity and insight. **2. Why This Question Matters Now**
In a digital age fueled by data-driven decisions, understanding probability underpins areas like finance, technology, and even personal finance planning. The growing interest in analytical thinking—fueled by awareness of randomness in algorithms, gaming odds, and forecasting—has sparked curiosity about handling structured randomness. This dice probability question taps into that momentum, offering a clear, tangible example of conditional likelihood. It’s relevant not just to educators and learners, but to anyone navigating environments where chance and pattern recognition intersect. **3. Breakdown: What’s the Probability?**
To calculate the chance of rolling exactly one 7 and no 10s with five 10-sided die rolls:
- Choose which one roll is the 7 – 5 possible positions
- That one roll must be 7 (probability 1/10)
- The other four rolls must be between 1–9 excluding 10 and not include 7 – nine acceptable values, so each has 9/10 chance
- Total probability = (5 choose 1) × (1/10) × (9/10)^4 = 5 × (1/10) × (6561/10000) = 32805 / 1000000
Approximately 0.0328 or 3.28% – a low but computable risk in structured randomness. **4. Real-World Applications**
This framework applies beyond dice: customer churn predictions, rollout timing in product launches, or statistical modeling of event sequences. Understanding how to isolate specific outcomes helps professionals assess rare but meaningful events, support innovation testing, and communicate uncertainty clearly—worthwhile skills in fields ranging from research to AI development.