Assume that a procedure yields a binomial distribution with a trial repeated 
n
=
14
 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of 
k
=
13
 successes given the probability 
q
=
0.29
 of success on a single trial.

(Report answer accurate to 4 decimal places.)

P
(
X
=
k
)
=
 


The probability of getting exactly \( k = 13 \) successes in a binomial distribution can be calculated using the binomial probability formula:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \),
- \( n = 14 \) is the number of trials,
- \( k = 13 \) is the number of successes,
- \( p = 0.29 \) is the probability of success on a single trial,
- \( q = 1 - p = 0.71 \) is the probability of failure on a single trial.

Now, let me calculate that for you.The probability of getting exactly 13 successes out of 14 trials, given a probability of success \( p = 0.29 \) for each trial, is approximately:

\[
P(X = 13) = 0.0000010
\]

Would you like further details on this calculation or have any questions?

Here are 5 related questions to consider:
1. How is the binomial coefficient \(\binom{n}{k}\) calculated?
2. What happens to the probability as the number of successes \( k \) approaches \( n \)?
3. How does the value of \( p \) affect the binomial distribution?
4. Can we use a normal approximation for binomial probabilities in this case?
5. How does the binomial distribution relate to real-world scenarios like pass/fail exams?

**Tip:** The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

Assume that a procedure yields a binomial distribution with a trial repeated n = 14 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k = 13 successes given the probability q = 0.29 of success on a single trial. (Report answer accurate to 4 decimal places.) P(X = k) = ?

Calculate the probability of getting 13 successes out of 14 trials in a binomial distribution with a success probability of 0.29. This problem covers the use of the binomial probability formula and is suitable for students in Grades 10-12.

Math Problem Statement

Assume that a procedure yields a binomial distribution with a trial repeated n

14 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k

13 successes given the probability q

P ( X

k )

Solution

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