About 59% of people who are murdered knew the person who committed the murder.

Suppose that Detective Smith's case load currently has 50 unsolved murders. Use a normal distribution approximation to the binomial distribution to find the following probabilities:

a. What is the probability that at least 26 of the victims among Detective Smith's case files knew their murderer?

b. What is the probability that between 26 and 33 of the victims knew their murderer?



To solve these problems, we’ll use the **normal approximation to the binomial distribution**. 

Let’s define the problem mathematically:

- \( n = 50 \) (number of unsolved murders)
- \( p = 0.59 \) (probability that the victim knew the murderer)
- \( q = 1 - p = 0.41 \) (probability that the victim did not know the murderer)

### Step 1: Compute the mean and standard deviation.
The **mean** \( \mu \) and **standard deviation** \( \sigma \) of a binomial distribution are given by:

\[
\mu = n \cdot p = 50 \cdot 0.59 = 29.5
\]

\[
\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{50 \cdot 0.59 \cdot 0.41} = \sqrt{12.095} \approx 3.48
\]

Now that we have the mean and standard deviation, we can use the **normal approximation to the binomial distribution**.

---

### Part (a): Probability that at least 26 of the victims knew their murderer.

We are looking for \( P(X \geq 26) \). In the normal approximation, the binomial random variable \( X \) is approximated by a normal distribution \( N(\mu, \sigma^2) \). To account for the discrete-to-continuous adjustment, we use a **continuity correction**:

\[
P(X \geq 26) \approx P(X \geq 25.5)
\]

Now, convert 25.5 to a **z-score** using the formula:

\[
z = \frac{X - \mu}{\sigma} = \frac{25.5 - 29.5}{3.48} = \frac{-4}{3.48} \approx -1.15
\]

Using the **z-table**, the cumulative probability for \( z = -1.15 \) is approximately 0.1251. Thus:

\[
P(X \geq 26) = 1 - P(Z < -1.15) = 1 - 0.1251 = 0.8749
\]

So, the probability that at least 26 of the victims knew their murderer is approximately:

\[
\boxed{0.8749}
\]

---

### Part (b): Probability that between 26 and 33 victims knew their murderer.

We are looking for \( P(26 \leq X \leq 33) \). With the continuity correction, this becomes:

\[
P(25.5 \leq X \leq 33.5)
\]

We now convert both limits to z-scores.

For \( X = 25.5 \):

\[
z = \frac{25.5 - 29.5}{3.48} = -1.15
\]

For \( X = 33.5 \):

\[
z = \frac{33.5 - 29.5}{3.48} = 1.15
\]

Using the **z-table**:
- \( P(Z < -1.15) \approx 0.1251 \)
- \( P(Z < 1.15) \approx 0.8749 \)

Now, the probability that \( 26 \leq X \leq 33 \) is:

\[
P(25.5 \leq X \leq 33.5) = P(Z < 1.15) - P(Z < -1.15) = 0.8749 - 0.1251 = 0.7498
\]

So, the probability that between 26 and 33 victims knew their murderer is approximately:

\[
\boxed{0.7498}
\]

---

### Summary of Results
- **Part (a):** \( P(X \geq 26) \approx 0.8749 \)
- **Part (b):** \( P(26 \leq X \leq 33) \approx 0.7498 \)

---

Do you need further details or have any follow-up questions?  

### 5 Related Questions:
1. What is the z-score for a given probability in a normal distribution?
2. How does continuity correction affect normal approximation?
3. When can we use the normal approximation for a binomial distribution?
4. How do we interpret z-scores and cumulative probabilities?
5. How would the probabilities change if the probability of knowing the murderer was 70%?

---

### Tip:
Always ensure that \( n \cdot p \) and \( n \cdot q \) are both greater than 5 when using the normal approximation to a binomial distribution for reliable results.

About 59% of people who are murdered knew the person who committed the murder. Suppose that Detective Smith's case load currently has 50 unsolved murders. Use a normal distribution approximation to the binomial distribution to find the following probabilities: a. What is the probability that at least 26 of the victims among Detective Smith's case files knew their murderer? b. What is the probability that between 26 and 33 of the victims knew their murderer?

Calculate the probability that at least 26 of the 50 victims in Detective Smith's unsolved murder cases knew their murderer using normal distribution approximation. This detailed solution explores key concepts like binomial distribution, continuity correction, and Z-scores. Suitable for students in Grades 11-12.

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