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This is a binomial probability problem, where we are given the probability of success (either a wolf is male or female) and the sample size. Let's break down the problem and solve it part by part.

### Part (a) Before 1918:
- The probability that a wolf is male: \( p = 0.60 \)
- The probability that a wolf is female: \( p = 0.40 \)
- Sample size: \( n = 10 \)

We will use the binomial distribution formula:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
Where:
- \( n \) is the number of trials (in this case, 10 wolves),
- \( k \) is the number of successes (males in this case),
- \( p \) is the probability of success.

For cumulative probabilities (like "7 or more"), we sum the probabilities of 7, 8, 9, and 10 wolves being male. To compute these efficiently, we use binomial cumulative distribution functions.

#### (a) What is the probability that **7 or more** wolves were male?

We need:
\[
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

#### (a) What is the probability that **fewer than 4** wolves were female?
Since the probability of being female is \( 0.40 \), this means:
\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]
where \( X \) is the number of female wolves.

### Part (b) After 1918:
- Probability that a wolf is male: \( p = 0.65 \)
- Probability that a wolf is female: \( p = 0.35 \)
- Sample size: \( n = 10 \)

#### (b) What is the probability that **7 or more** wolves were male?

Again, we will calculate:
\[
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

#### (b) What is the probability that **fewer than 4** wolves were female?

We need:
\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

I will now calculate the answers for you.Here are the calculated probabilities:

### (a) Before 1918:
- The probability that **7 or more** wolves were male: \( 0.382 \)
- The probability that **fewer than 4** wolves were female: \( 0.382 \)

### (b) After 1918:
- The probability that **7 or more** wolves were male: \( 0.514 \)
- The probability that **fewer than 4** wolves were female: \( 0.514 \)

Would you like further details or clarifications on these calculations?

Here are five related questions for further exploration:
1. How would the probabilities change if the sample size increased to 20 wolves?
2. What if the percentage of males before 1918 was 70% instead of 60%?
3. How do these binomial probabilities relate to the normal approximation?
4. How can the binomial theorem be used in other wildlife population studies?
5. What is the significance of the symmetry in the probabilities for males and females?

**Tip:** The binomial distribution is great for modeling events with two outcomes, but for large \( n \), consider using a normal approximation for faster calculations.

Before 1918, approximately 60% of the wolves in a region were male, and 40% were female. After 1918, the male proportion increased to 65%, and the female proportion dropped to 35%. In a sample of 10 wolves, the task is to calculate the probabilities for the following cases: (a) before 1918, (b) after 1918. Specifically, we need: (1) The probability that 7 or more wolves are male. (2) The probability that fewer than 4 wolves are female.

This problem explores binomial probabilities of wolves in a region, given shifting male-to-female ratios before and after 1918. We calculate the probabilities of having 7 or more males and fewer than 4 females in a sample of 10 wolves, using the binomial distribution.

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