In a binomial distribution, the probability of $k$ successes in $n$ trials is given by the formula:

$P(X = k) = \binom{n}{k} p^k q^{n-k}$

where:

• $n$ is the number of trials,
• $k$ is the number of successes,
• $p$ is the probability of success on a single trial, and
• $q$ is the probability of failure on a single trial ($q = 1 - p$).

In this case:

• $n = 14$,
• $k = 0$,
• $q = 0.71$.

Since $q$ is the probability of failure, the probability of success, $p$, is:

$p = 1 - q = 1 - 0.71 = 0.29$

Now, substitute the values into the formula for $P(X = 0)$:

$P(X = 0) = \binom{14}{0} (0.29)^0 (0.71)^{14}$

Calculating this:

$\binom{14}{0} = 1 \quad \text{(since } \binom{n}{0} = 1 \text{ for any } n\text{)}$ $(0.29)^0 = 1 \quad \text{(since any number to the power of 0 is 1)}$ $(0.71)^{14}$

So,

$P(X = 0) = 1 \times 1 \times (0.71)^{14}$

Now let's compute $(0.71)^{14}$.The probability of having 0 successes (all trials failing) is approximately $0.0083$ (rounded to four decimal places).

Do you want details or have any questions?

Here are some related questions to consider:

1. What is the probability of having exactly 1 success in 14 trials?
2. How would the probability change if the probability of failure, $q$, was 0.5 instead of 0.71?
3. What is the probability of having at least one success in 14 trials?
4. How does the expected number of successes change if the number of trials $n$ is doubled?
5. How would you use Excel or a statistical software to calculate these probabilities?

Tip: For quick probability calculations, using software tools like Excel or a graphing calculator can save time and reduce errors.