Suppose n=15 and r=4.
What is the probability of x=3  for  n=10 (to 4 decimals)?

We are dealing with a binomial probability question where the formula for the binomial probability \( P(x=k) \) is given by:

\[
P(x = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( n \) is the number of trials,
- \( k \) is the number of successes (in this case, \( x = 3 \)),
- \( p \) is the probability of success,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Here, \( n = 10 \), \( x = 3 \), and we need to find \( p \). 

The probability \( p \) is often given as \( \frac{r}{n} \). So, using \( n = 15 \) and \( r = 4 \), the probability \( p \) will be:

\[
p = \frac{r}{n} = \frac{4}{15}
\]

Now we can calculate the binomial probability \( P(x = 3) \) for \( n = 10 \), \( x = 3 \), and \( p = \frac{4}{15} \).

Let me calculate that for you.The probability of \( x = 3 \) when \( n = 10 \) and \( p = \frac{4}{15} \) is approximately 0.2595.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How is the binomial coefficient calculated?
2. What is the meaning of a binomial distribution in probability?
3. How does changing \( p \) affect the binomial probability?
4. How can we use binomial probabilities to model real-life scenarios?
5. What is the difference between a binomial and a normal distribution?

**Tip:** Binomial probabilities are useful when modeling events with only two outcomes, like success or failure.

Suppose n=15 and r=4. What is the probability of x=3 for n=10 (to 4 decimals)?

Discover how to calculate the binomial probability of x=3 when n=10 and the probability of success is 4/15. This solution provides a detailed breakdown of the binomial formula, probability calculation, and essential concepts in combinatorics and probability theory. Ideal for students in Grades 10-12.

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