A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.
n equals 9​, p equals 0.5​, x less than or equals 3
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Part 1
The probability of x less than or equals 3 successes is 
  
enter your response here. ​(Round to four decimal places as​ needed.)

In a binomial probability experiment, the probability of having exactly \( x \) successes in \( n \) independent trials is given by the binomial probability formula:

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

Where:
- \( n \) is the number of trials,
- \( p \) is the probability of success on a single trial,
- \( x \) is the number of successes,
- \( \binom{n}{x} \) is the binomial coefficient.

You are asked to find the probability of getting \( x \leq 3 \) successes when \( n = 9 \) and \( p = 0.5 \). This means we need to sum the probabilities for \( x = 0, 1, 2, 3 \):

\[
P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

Let’s calculate each of these probabilities:

### For \( x = 0 \):
\[
P(X = 0) = \binom{9}{0} \cdot 0.5^0 \cdot (1 - 0.5)^9 = 1 \cdot 1 \cdot 0.001953125 = 0.001953125
\]

### For \( x = 1 \):
\[
P(X = 1) = \binom{9}{1} \cdot 0.5^1 \cdot (1 - 0.5)^8 = 9 \cdot 0.5 \cdot 0.00390625 = 0.017578125
\]

### For \( x = 2 \):
\[
P(X = 2) = \binom{9}{2} \cdot 0.5^2 \cdot (1 - 0.5)^7 = 36 \cdot 0.25 \cdot 0.0078125 = 0.0703125
\]

### For \( x = 3 \):
\[
P(X = 3) = \binom{9}{3} \cdot 0.5^3 \cdot (1 - 0.5)^6 = 84 \cdot 0.125 \cdot 0.015625 = 0.1640625
\]

Now, sum all these probabilities:

\[
P(X \leq 3) = 0.001953125 + 0.017578125 + 0.0703125 + 0.1640625 = 0.25390625
\]

Thus, the probability of \( x \leq 3 \) successes is approximately:

\[
\boxed{0.2539}
\]

Would you like more details or have any questions on this solution?

Here are 5 related questions to deepen your understanding:
1. How do you compute binomial coefficients in the probability formula?
2. What happens to binomial probabilities if \( p \) is different from 0.5?
3. How can the binomial probability formula be used for large \( n \)?
4. What is the cumulative distribution function in binomial probability?
5. How do we approximate binomial probabilities using the normal distribution?

**Tip**: Always check if a problem asks for "less than or equal to" (\( \leq \)) or "exactly" for correct interpretation of cumulative probabilities.

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n equals 9, p equals 0.5, x less than or equals 3. The probability of x less than or equals 3 successes is ​(Round to four decimal places as​ needed.)

Calculate the binomial probability of getting x ≤ 3 successes in 9 independent trials with a success probability of 0.5. This solution involves binomial distribution formulas and cumulative probability. Suitable for students in Grades 9-12.

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