Thuy Rolls A Number Cube 7 Times. Which Expression Represents The Probability Of Rolling A 4 Exactly 2 Times? P ( K Successes ) = ( N K ) P K ( 1 − P ) N − K P(k \text{ Successes }) = \binom{n}{k} P^k (1-P)^{n-k} P ( K Successes ) = ( K N ) P K ( 1 − P ) N − K Where:- { \binom{n}{k} = \frac{n!}{(n-k)! \cdot K!} $}$For This
Introduction
Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. In this article, we will explore the probability of rolling a 4 exactly 2 times when Thuy rolls a number cube 7 times. We will use the binomial probability formula to calculate the probability of this event.
The Binomial Probability Formula
The binomial probability formula is given by:
P(k successes) = (n choose k) * P^k * (1-P)^(n-k)
where:
- P(k successes) is the probability of k successes
- (n choose k) is the number of combinations of n items taken k at a time
- P is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
Calculating the Probability of Rolling a 4 Exactly 2 Times
In this case, we want to find the probability of rolling a 4 exactly 2 times when Thuy rolls a number cube 7 times. We can use the binomial probability formula to calculate this probability.
First, we need to determine the probability of success on a single trial. Since there are 6 possible outcomes when rolling a number cube (1, 2, 3, 4, 5, and 6), the probability of rolling a 4 on a single trial is:
P = 1/6
Next, we need to determine the number of trials (n) and the number of successes (k). In this case, n = 7 (since Thuy rolls the cube 7 times) and k = 2 (since we want to find the probability of rolling a 4 exactly 2 times).
Now, we can plug these values into the binomial probability formula:
P(2 successes) = (7 choose 2) * (1/6)^2 * (5/6)^5
Calculating the Number of Combinations
The number of combinations of n items taken k at a time is given by:
(n choose k) = n! / ((n-k)! * k!)
In this case, we need to calculate (7 choose 2). We can do this using the formula above:
(7 choose 2) = 7! / ((7-2)! * 2!) = 7! / (5! * 2!) = (7 * 6 * 5!) / (5! * 2!) = (7 * 6) / 2 = 21
Calculating the Probability
Now that we have calculated the number of combinations, we can plug this value into the binomial probability formula:
P(2 successes) = 21 * (1/6)^2 * (5/6)^5
To calculate this probability, we need to evaluate the expression:
(1/6)^2 = 1/36 (5/6)^5 = 0.1615 (approximately)
Now, we can multiply these values together:
P(2 successes) = 21 * (1/36) * 0.1615 = 0.1025 (approximately)
Conclusion
In this article, we used the binomial probability formula to calculate the probability of rolling a 4 exactly 2 times when Thuy rolls a number cube 7 times. We determined the probability of success on a single trial, calculated the number of combinations, and evaluated the binomial probability formula to find the probability of rolling a 4 exactly 2 times.
The probability of rolling a 4 exactly 2 times is approximately 0.1025. This means that if Thuy rolls the cube 7 times, the probability of rolling a 4 exactly 2 times is approximately 10.25%.
References
- "Probability" by Khan Academy
- "Binomial Probability Formula" by Math Is Fun
- "Combinations" by Math Is Fun
Discussion
What do you think about the binomial probability formula? Have you ever used it to calculate a probability? Share your thoughts and experiences in the comments below!
Related Topics
- Probability
- Binomial Probability Formula
- Combinations
- Permutations
- Statistics
Further Reading
- "Probability and Statistics" by Khan Academy
- "Binomial Probability" by Math Is Fun
- "Combinations and Permutations" by Math Is Fun
Introduction
In our previous article, we explored the probability of rolling a 4 exactly 2 times when Thuy rolls a number cube 7 times. We used the binomial probability formula to calculate this probability. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the binomial probability formula?
A: The binomial probability formula is a mathematical formula used to calculate the probability of k successes in n independent trials, where the probability of success on a single trial is p. The formula is given by:
P(k successes) = (n choose k) * p^k * (1-p)^(n-k)
Q: What is the meaning of (n choose k)?
A: (n choose k) is the number of combinations of n items taken k at a time. It is calculated using the formula:
(n choose k) = n! / ((n-k)! * k!)
Q: How do I calculate the number of combinations?
A: To calculate the number of combinations, you can use the formula:
(n choose k) = n! / ((n-k)! * k!)
For example, if you want to calculate (7 choose 2), you can use the formula:
(7 choose 2) = 7! / ((7-2)! * 2!) = 7! / (5! * 2!) = (7 * 6 * 5!) / (5! * 2!) = (7 * 6) / 2 = 21
Q: What is the probability of success on a single trial?
A: The probability of success on a single trial is the probability of rolling a 4 on a single roll of the number cube. Since there are 6 possible outcomes (1, 2, 3, 4, 5, and 6), the probability of rolling a 4 on a single trial is:
P = 1/6
Q: How do I calculate the probability of rolling a 4 exactly 2 times?
A: To calculate the probability of rolling a 4 exactly 2 times, you can use the binomial probability formula:
P(2 successes) = (7 choose 2) * (1/6)^2 * (5/6)^5
Q: What is the probability of rolling a 4 exactly 2 times?
A: The probability of rolling a 4 exactly 2 times is approximately 0.1025.
Q: Can I use the binomial probability formula to calculate the probability of rolling a 4 more than 2 times?
A: Yes, you can use the binomial probability formula to calculate the probability of rolling a 4 more than 2 times. However, you will need to calculate the probability of rolling a 4 3, 4, 5, 6, or 7 times.
Q: Can I use the binomial probability formula to calculate the probability of rolling a 4 less than 2 times?
A: Yes, you can use the binomial probability formula to calculate the probability of rolling a 4 less than 2 times. However, you will need to calculate the probability of rolling a 4 0, 1, or 2 times.
Conclusion
In this article, we answered some frequently asked questions related to the binomial probability formula and the probability of rolling a 4 exactly 2 times when Thuy rolls a number cube 7 times. We hope that this article has been helpful in understanding the binomial probability formula and its applications.
References
- "Probability" by Khan Academy
- "Binomial Probability Formula" by Math Is Fun
- "Combinations" by Math Is Fun
Discussion
Do you have any questions about the binomial probability formula or the probability of rolling a 4 exactly 2 times? Share your thoughts and experiences in the comments below!
Related Topics
- Probability
- Binomial Probability Formula
- Combinations
- Permutations
- Statistics
Further Reading
- "Probability and Statistics" by Khan Academy
- "Binomial Probability" by Math Is Fun
- "Combinations and Permutations" by Math Is Fun