To solve this problem, we can use the binomial probability formula, since each person has only two possible outcomes: either they are in their twenties or they are not. The probability of someone being in their twenties is 19% (or 0.19), and the probability of them not being in their twenties is 81% (or 0.81). We are asked to find the probability that exactly two or three people in a group of eight are in their twenties.

Binomial Probability Formula:

The binomial probability formula is given by:

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

• $P(X = k)$ is the probability of exactly $k$ successes (in this case, people in their twenties).
• $n$ is the number of trials (8 people in the group).
• $k$ is the number of successes (2 or 3 people in their twenties).
• $p$ is the probability of success (0.19).
• $1 - p$ is the probability of failure (0.81).
• $\binom{n}{k}$ is the binomial coefficient, which can be calculated as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Step 1: Find $P(X = 2)$

Using the binomial formula:

$P(X = 2) = \binom{8}{2} (0.19)^2 (0.81)^6$

First, calculate the binomial coefficient $\binom{8}{2}$:

$\binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28$

Now calculate the rest:

$P(X = 2) = 28 \times (0.19)^2 \times (0.81)^6$ $P(X = 2) = 28 \times 0.0361 \times 0.2621 \approx 28 \times 0.00946 \approx 0.26488$

Step 2: Find $P(X = 3)$

Using the binomial formula again:

$P(X = 3) = \binom{8}{3} (0.19)^3 (0.81)^5$

First, calculate the binomial coefficient $\binom{8}{3}$:

$\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

Now calculate the rest:

$P(X = 3) = 56 \times (0.19)^3 \times (0.81)^5$ $P(X = 3) = 56 \times 0.006859 \times 0.3302 \approx 56 \times 0.002267 \approx 0.12697$

Step 3: Find the total probability for 2 or 3 people in their twenties

Now, sum the probabilities for $P(X = 2)$ and $P(X = 3)$:

$P(X = 2 \text{ or } X = 3) = P(X = 2) + P(X = 3)$ $P(X = 2 \text{ or } X = 3) = 0.26488 + 0.12697 = 0.39185$

Final Answer:

The probability that two or three of the people in the group are in their twenties is approximately 0.3919, or 39.19%.

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Would you like further details on any part of this solution?

Here are some related questions to explore:

1. How would the probability change if the group size increased to 10 people?
2. How do you compute the expected number of people in their twenties in a group of 8?
3. What happens to the binomial probability when the probability of success (being in their twenties) increases to 50%?
4. How can you calculate the cumulative probability of having fewer than 3 people in their twenties?
5. What is the variance of the binomial distribution in this problem?

Tip: When working with binomial probabilities, it's often helpful to use a calculator or software to compute binomial coefficients and powers for efficiency.