Part 1:

We are given that 37% (or 0.37) of people in a certain area own a dog. The problem asks us to find the probability that exactly 6 out of 10 randomly selected people own a dog.

This is a binomial probability problem, where the probability of success (owning a dog) is $p = 0.37$, the number of trials is $n = 10$, and the number of successes (people owning a dog) is $x = 6$.

The binomial probability formula is:

$P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}$

Where:

• $\binom{n}{x}$ is the binomial coefficient (combination), which is calculated as $\frac{n!}{x!(n - x)!}$,
• $p = 0.37$ is the probability of owning a dog,
• $n = 10$ is the number of people,
• $x = 6$ is the number of successes (people owning a dog).

Let's compute the binomial probability:

$P(X = 6) = \binom{10}{6} (0.37)^6 (0.63)^4$

Using the formula for the binomial coefficient:

$\binom{10}{6} = \frac{10!}{6!(10 - 6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$

Now calculate:

$P(X = 6) = 210 \times (0.37)^6 \times (0.63)^4$

Using a calculator:

$P(X = 6) \approx 0.085$

So, the probability that exactly 6 out of 10 people own a dog is approximately 0.085.

Part 2:

Now, we need to find the probability that 66 or fewer out of 10 randomly selected people own a dog. This is the cumulative probability for $X \leq 6$, meaning we need to sum the probabilities of $X = 0, 1, 2, \dots, 6$.

This can be calculated using the binomial cumulative distribution function (CDF):

$P(X \leq 6) = P(X = 0) + P(X = 1) + \dots + P(X = 6)$

The cumulative probability can be calculated using statistical software or a binomial calculator. Using this method (or a standard calculator with a CDF function), we find:

$P(X \leq 6) \approx 0.914$

So, the probability that 66 or fewer out of 10 randomly selected people own a dog is approximately 0.914.

Final Answer:

• Part 1: $P(X = 6) \approx 0.085$
• Part 2: $P(X \leq 6) \approx 0.914$

Would you like further clarification or details on the steps involved?

Here are 5 related questions:

1. What is the binomial distribution formula and when is it used?
2. How do you calculate cumulative probabilities for a binomial distribution?
3. What are the differences between binomial and normal distributions?
4. How do you find the mean and variance for a binomial distribution?
5. Can the binomial distribution be approximated using a normal distribution?

Tip: When dealing with binomial probabilities, you can use a binomial calculator or statistical software (like Excel or a graphing calculator) to save time on computing the individual terms.