Question: If I roll a fair six-sided die four times, what is the probability that I roll the number 4 exactly twice? - IQnection
Probability of Rolling Exactly Two 4s When Rolling a Die Four Times
Probability of Rolling Exactly Two 4s When Rolling a Die Four Times
Rolling a fair six-sided die four times is a classic probability scenario that many people encounter, whether in games, education, or just casual curiosity. A common question arises: If I roll a fair six-sided die four times, what is the probability that I roll the number 4 exactly twice? Understanding this probability involves applying the principles of binomial probability, making it a great example to explore how random events and combinations work.
Understanding the Binomial Probability Framework
Understanding the Context
This problem fits perfectly within the binomial distribution framework. The binomial distribution applies when:
- There are a fixed number of independent trials (here, 4 die rolls).
- Each trial has only two outcomes: âÃÂÃÂsuccessâÃÂà(rolling a 4) or âÃÂÃÂfailureâÃÂà(rolling anything other than 4).
- The probability of success remains constant per trial (for a fair die, P(4) = 1/6).
- Trials are independent.
In this context:
- Success = rolling a 4 (probability ( p = rac{1}{6} ))
- Failure = rolling not a 4 (probability ( q = 1 - p = rac{5}{6} ))
- Number of trials ( n = 4 )
- Desired number of successes ( k = 2 )
Step-by-Step Calculation of the Probability
Image Gallery
Key Insights
1. Calculate the number of favorable outcomes
We need the number of ways to roll exactly two 4s in four rolls. This is a combination problem:
[
inom{4}{2} = rac{4!}{2!(4-2)!} = rac{24}{2 \cdot 2} = 6
]
There are 6 unique sequences (e.g., 4,4,n,n in all combinations) where exactly two rolls show a 4.
2. Calculate the probability for one such sequence
🔗 Related Articles You Might Like:
📰 Click, Treat, Repeat! Dog Clicker Games Are Taking Training to a Whole New Level 📰 Dog Express Merge & Puzzle: The Ultimate Challenge Thatll Blow Your Mind! 📰 Shocking Dog Express Merge & Puzzle Quest: The Puzzle Thats Taken the Internet by Storm! 📰 Orgins 9982347 📰 Alcombras Untold Story Threatens To Change Everything You Thought You Knew 1726031 📰 Hawaii From Los Angeles Flight Time 4382315 📰 Finally Miracast Iphone Secrets To Stream Any Device Instantlyshocking Results 6044215 📰 Studio Lite 3426320 📰 Unlock Fidelity Retirements Power Top Strategies To Grow Your Savings Fast 8902927 📰 Pe Ratio Meaning 3080276 📰 Watch Party Pacers 356801 📰 Apple Car Play Adapter 5005178 📰 Clp Us Explained What You Need To Know Before It Affects Your Business 6923747 📰 Master Oracle Mrp Planning Like A Proinside This Game Changing Strategy 8817009 📰 How To Make Money With Money 3661823 📰 Ftw With Sata Mtta Discover The Game Changing Storage Hacks Today 3757492 📰 Fire Emblem 16 4943734 📰 How A Single Dollar Became A Financial Alarm 8046708Final Thoughts
For any specific sequence with exactly two 4s and two non-4s (e.g., 4, 4, 2, 5), the probability is:
[
P = \left(rac{1}{6}
ight)^2 \ imes \left(rac{5}{6}
ight)^2 = rac{1}{36} \ imes rac{25}{36} = rac{25}{1296}
]
3. Multiply by the number of favorable sequences
Since the 6 arrangements are mutually exclusive, the total probability is:
[
P(\ ext{exactly 2 fours}) = inom{4}{2} \ imes \left(rac{1}{6}
ight)^2 \ imes \left(rac{5}{6}
ight)^2 = 6 \ imes rac{25}{1296} = rac{150}{1296}
]
4. Simplify the result
[
rac{150}{1296} = rac{25}{216} pprox 0.1157 \ ext{ or } 11.57%
]
Final Answer
The probability of rolling exactly two 4s when rolling a fair six-sided die four times is:
[
oxed{rac{25}{216}} \quad \ ext{or approximately} \quad 11.57%
]