This problem is a classic example of a binomial probability problem, where we have two possible outcomes:

• A person believes civil rights for Black Americans have improved during their lifetime (success).
• A person does not believe so (failure).

We are given:

• The probability of success ($p$) is 59% or 0.59.
• The probability of failure ($q$) is 1 - 0.59 = 0.41.
• The number of trials ($n$) is 10 (since 10 US adults are randomly selected).
• We need to find the probability that exactly 2 out of the 10 people think civil rights for Black Americans have improved, which means we are looking for $P(X = 2)$, where $X$ is the number of successes.

The formula for binomial probability is:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$,
• $p$ is the probability of success,
• $k$ is the number of successes (2 in this case),
• $n$ is the number of trials (10 in this case).

Step 1: Calculate the binomial coefficient $\binom{10}{2}$

$\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45$

Step 2: Calculate the probability

Now, plug everything into the binomial probability formula:

$P(X = 2) = 45 \times (0.59)^2 \times (0.41)^8$

Step 3: Simplify and calculate the result

Let's calculate this:

$(0.59)^2 = 0.3481$ $(0.41)^8 = 0.0001069$ Now multiply all terms together:

$P(X = 2) = 45 \times 0.3481 \times 0.0001069 \approx 0.001678$

So, the probability that exactly 2 out of the 10 people believe civil rights for Black Americans have improved during their lifetime is approximately 0.0017 or 0.17%.

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Would you like further details or have any additional questions? Here are some related questions that might interest you:

1. What happens to the probability if we increase the number of people selected?
2. How do we interpret the probability when we have such a small value?
3. Can you explain the concept of binomial distribution in more detail?
4. What would be the probability if we wanted to know the likelihood that exactly 5 people believe civil rights have improved?
5. How does the probability change if the belief rate changes to 70% instead of 59%?

Tip: The binomial distribution is most useful when there are a fixed number of independent trials, each with only two possible outcomes, and the probability of success remains constant across trials.