Senem is auditing the tax returns of individuals in three zip codes. Thus far, $$ 93.1 \% $$ of the audited tax returns have passed the audit. If Senem takes 15 tax returns, what is the probability that at most 14 returns will pass the audit? - Round your answer to 3 decimal places.

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To determine the probability that at most 14 out of 15 tax returns will pass the audit, we can use the **binomial probability** model. Here's a step-by-step breakdown:

1. **Parameters:**
   - **Probability of success (passing the audit), $$ p $$:** 93.1% or 0.931
   - **Number of trials, $$ n $$:** 15
   - **Number of successes, $$ k $$:** At most 14

2. **Understanding "At Most 14":**
   - This means we need to calculate the probability of having **0 to 14** passes.
   - Mathematically, this is $$ P(X \leq 14) $$.

3. **Complement Rule:**
   - Instead of calculating $$ P(X \leq 14) $$ directly, it's easier to use the complement rule:
   $$
   P(X \leq 14) = 1 - P(X = 15)
   $$
   - Here, $$ P(X = 15) $$ is the probability that **all 15** tax returns pass the audit.

4. **Calculating $$ P(X = 15) $$:**
   - Using the binomial probability formula:
   $$
   P(X = 15) = p^{15} = 0.931^{15}
   $$
   - Calculating $$ 0.931^{15} $$ gives approximately **0.342**.

5. **Final Calculation:**
   - Substituting back:
   $$
   P(X \leq 14) = 1 - 0.342 = 0.658
   $$

**Rounded to three decimal places, the probability is 0.658.**

**Answer:** 0.658
The probability that at most 14 out of 15 tax returns will pass the audit is 0.658.

Senem is auditing the tax returns of individuals in three zip codes. Thus far, \( 93.1 \% \) of the audited tax returns have passed the audit. If Senem takes 15 tax returns, what is the probability that at most 14 returns will pass the audit? - Round your answer to 3 decimal places.

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