Assume that a procedure yields a binomial distribution with a trial repeated n=9 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k=1  successes given the probability q=0.61 of failure on a single trial.

(Report answer accurate to 4 decimal places.)

 

The binomial probability formula is:

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

Where:
- \( n = 9 \) (number of trials)
- \( k = 1 \) (number of successes)
- \( p = 1 - q = 1 - 0.61 = 0.39 \) (probability of success on a single trial)
- \( q = 0.61 \) (probability of failure on a single trial)

We need to calculate the probability of getting exactly 1 success:

1. Calculate \( \binom{n}{k} = \binom{9}{1} = 9 \)
2. Substitute the values into the binomial formula:

\[
P(X = 1) = \binom{9}{1} (0.39)^1 (0.61)^{9-1}
\]

Let's compute this now.The probability of getting exactly 1 success out of 9 trials, with a probability of failure \( q = 0.61 \), is approximately \( 0.0673 \) (accurate to 4 decimal places).

Would you like further details or clarification?

Here are five related questions for deeper understanding:
1. How does the probability change if the number of trials increases?
2. What happens to the probability if the success rate \( p \) increases?
3. Can you explain how to derive the binomial coefficient?
4. How would you calculate cumulative probabilities (e.g., \( P(X \leq 1) \)) in a binomial distribution?
5. How does a binomial distribution differ from a normal distribution?

**Tip:** For large \( n \), the binomial distribution can be approximated by a normal distribution when certain conditions hold (like \( np \) and \( nq \) both being sufficiently large).

Assume that a procedure yields a binomial distribution with a trial repeated n=9 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k=1 successes given the probability q=0.61 of failure on a single trial. (Report answer accurate to 4 decimal places.)

Learn how to calculate the probability of exactly 1 success in 9 trials using the binomial probability formula. This problem explores key probability concepts, such as binomial distribution and the calculation of success and failure rates. Ideal for students in Grades 10-12.

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