Assume that a procedure yields a binomial distribution with a trial repeated 
 n=8 times. Use either the binomial probability formula (or technology) to find the probability of 
 k=1 successes given the p=.34 probability 
 of success on a single trial.



To find the probability of getting exactly \( k = 1 \) success in \( n = 8 \) trials of a binomial experiment with a probability of success \( p = 0.34 \) on a single trial, we use the **binomial probability formula**:

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

Where:
- \( n = 8 \) (number of trials),
- \( k = 1 \) (number of successes),
- \( p = 0.34 \) (probability of success on a single trial),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

Let's calculate step by step:

1. **Binomial coefficient** \( \binom{8}{1} = \frac{8!}{1!(8-1)!} = 8 \)
2. **Success term** \( p^k = (0.34)^1 = 0.34 \)
3. **Failure term** \( (1 - p)^{n-k} = (1 - 0.34)^{8-1} = (0.66)^7 \)

Now calculate the entire expression:

\[
P(X = 1) = 8 \times (0.34) \times (0.66)^7
\]

Let me compute that for you.The probability of getting exactly \( k = 1 \) success in \( n = 8 \) trials with a probability of success \( p = 0.34 \) on each trial is approximately \( 0.148 \), or 14.8%.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How does changing the number of trials \( n \) affect the binomial probability?
2. What is the probability of getting exactly \( k = 0 \) successes?
3. How do you calculate the cumulative probability for \( k \leq 1 \) successes?
4. What happens to the binomial distribution if \( p \) increases to 0.5?
5. How would you interpret the results if the probability of success was very close to 1?

**Tip:** Binomial probabilities add up to 1 for all possible outcomes \( k \) from 0 to \( n \).

Assume that a procedure yields a binomial distribution with a trial repeated n=8 times. Use either the binomial probability formula (or technology) to find the probability of k=1 successes given the p=.34 probability of success on a single trial.

Learn how to calculate the probability of exactly 1 success in 8 trials using the binomial probability formula, given a success probability of 0.34 per trial. This problem explores binomial distribution and provides a detailed step-by-step solution. Suitable for Grades 10-12.

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