Before 1918 in a random sample of 10 wolves spotted in the region what is the probability that 7 or more were male?

To calculate the probability that 7 or more wolves out of a sample of 10 are male, we need to make an assumption about the probability that any given wolf is male. For simplicity, let's assume that the probability of a wolf being male is 0.5 (i.e., a 50/50 chance).

This scenario follows a **binomial distribution** where:
- \( n = 10 \): the number of trials (wolves),
- \( k \geq 7 \): the number of successful trials (male wolves),
- \( p = 0.5 \): the probability of each trial being a success (male wolf).

We want to find \( P(X \geq 7) \) where \( X \sim \text{Binomial}(10, 0.5) \).

### Step 1: Define the Probability Formula
For a binomial distribution, the probability of getting exactly \( k \) successes in \( n \) trials is given by:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
where \( \binom{n}{k} \) is the binomial coefficient, \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \).

### Step 2: Calculate \( P(X \geq 7) \)
Since we want 7 or more male wolves, we need to calculate:
\[
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

### Step 3: Apply the Formula for Each Case
1. **For \( X = 7 \):**
   \[
   P(X = 7) = \binom{10}{7} (0.5)^7 (0.5)^3
   \]

2. **For \( X = 8 \):**
   \[
   P(X = 8) = \binom{10}{8} (0.5)^8 (0.5)^2
   \]

3. **For \( X = 9 \):**
   \[
   P(X = 9) = \binom{10}{9} (0.5)^9 (0.5)^1
   \]

4. **For \( X = 10 \):**
   \[
   P(X = 10) = \binom{10}{10} (0.5)^{10}
   \]

### Step 4: Sum the Probabilities
Once each of these probabilities is calculated, summing them will give the total probability of observing 7 or more male wolves in a sample of 10.

Let me know if you'd like the exact decimal answer calculated, or have questions on any steps so far!

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Would you like details on the full calculation?

### Related Questions

1. How would the probability change if the chance of any given wolf being male were 0.6 instead?
2. What is the probability of exactly 5 wolves being male in a sample of 10?
3. If we wanted at least 8 wolves to be male, how would that affect the probability?
4. How does binomial probability differ from normal approximation methods?
5. How would this probability be computed for a larger sample, say 50 wolves?

#### Tip
For binomial distributions with large sample sizes, the normal approximation can sometimes simplify calculations for probabilities over a range of values.

Explore the binomial probability calculation for finding the likelihood that 7 or more wolves in a random sample of 10 are male, assuming a 50% chance for each wolf. This detailed solution uses key probability concepts and binomial distribution formulas. Suitable for high school students in Grades 10-12.

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