To solve this problem, we need to determine the probability that 8 or more wolves in a random sample of 11 are male, given that 55% of the wolves in the region were male before 1918.

This situation can be modeled using the binomial distribution, which is used when we have a fixed number of trials (in this case, 11 wolves), each trial has two possible outcomes (male or female), and the probability of success (a male wolf) is constant across trials.

The binomial probability formula is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ Where:

• $n = 11$ (the number of trials, i.e., wolves),
• $k$ is the number of successes (male wolves),
• $p = 0.55$ (the probability of a male wolf),
• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

We need to find the probability that 8 or more wolves are male:

$P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11)$

Let's compute each of these probabilities.

We can calculate these using the binomial distribution formula:

$P(X = k) = \binom{11}{k} (0.55)^k (0.45)^{11-k}$

I'll compute this now. math

Parameters

n = 11 # number of trials (wolves spotted) p = 0.55 # probability of male wolf q = 1 - p # probability of female wolf

Function to calculate binomial probability

def binomial_prob(n, k, p): return math.comb(n, k) * (pk) * (q(n-k))

Calculate P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11)

prob_8_or_more = sum(binomial_prob(n, k, p) for k in range(8, 12))

round(prob_8_or_more, 3)The probability that 8 or more wolves in a random sample of 11 are male, given that 55% of wolves in the region were male before 1918, is approximately 0.191.

Would you like more details on the calculations or further questions? Here are a few related questions for exploration:

1. How would the probability change if the percentage of male wolves before 1918 was 60% instead of 55%?
2. How does the binomial distribution model compare to the normal distribution for large sample sizes?
3. What are the expected number of male wolves in a sample of 11, given a 55% probability of a male wolf?
4. If 7 or fewer wolves are male, what is the probability of that happening in this scenario?
5. How would the probability be affected if the sample size was increased to 20 wolves?

Tip: For problems like this, it's helpful to use a binomial cumulative distribution calculator to quickly compute probabilities for a range of outcomes.