Question: If I roll a fair six-sided die four times, what is the probability that I roll the number 4 exactly twice? - Coaching Toolbox
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:
📰 government shutdown march 2025 📰 grosjean 📰 how many tablespoons is a cup 📰 Applied Materials Inc 2974663 📰 New Car Calculator 871730 📰 Appleseed Movie 7280644 📰 Linear Fit Excel 8791667 📰 Chinese Camp 5872250 📰 Unlock Faster Results Master Sales And Operations Planning Sop Today 3066316 📰 New Years Eve Indianapolis 2024 9459850 📰 Hide Behind This Batman Maskunleash Your Inner Hero Now 9665148 📰 Headache And Diarrhoea 5523212 📰 Pink Glock Unveiled The Stunning Design Thats Taking Firearms By Storm 2156315 📰 End Of Cold War 933630 📰 Carrie Underwood Jesus Take 8144822 📰 Citizenm Boston 2994331 📰 Diplomats Back Minsk Process Experts Believe Fresh Talks Are Key To Lasting Ceasefire 508671 📰 The Shocking Truth About Slyward Youll Never Guess His Past 9481777Final 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%
]