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A process with a binomial distribution has a 5% probability of success. What is the approximate probability of having zero successes if 10 samples are drawn from the process?

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To determine the probability of having zero successes in a binomial distribution, the relevant formula is:

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

where:

- \( P(X = k) \) is the probability of k successes in n trials,

- \( n \) is the number of trials (in this case, 10),

- \( k \) is the number of successes (here, 0),

- \( p \) is the probability of success on a single trial (5% or 0.05), and

- \( (1 - p) \) is the probability of failure on a single trial (95% or 0.95).

For zero successes (k = 0), the formula simplifies to:

\[ P(X = 0) = \binom{10}{0} (0.05)^0 (0.95)^{10} \]

\[ P(X = 0) = 1 \times 1 \times (0.95)^{10} \]

Now, calculating \( (0.95)^{10} \):

Using the approximation:

\[ (0.95

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