In one town 76% of adults have health insurance, find the probability that 8 adults selected at random from the town all have health insurance. would it be unusual to select 8 people and find they all have health insurance. and why?

0.1113

To find the probability that 8 adults selected at random from the town all have health insurance, we can use the concept of independent events.

Since each person's health insurance status is independent of others, the probability of each individual having health insurance is 76%.

Now, we can calculate the probability of all 8 selected adults having health insurance by multiplying the probabilities for each individual having health insurance:

P(8 adults have health insurance) = 0.76^8 ≈ 0.1001

The probability of selecting 8 adults at random from the town and finding that they all have health insurance is approximately 0.1001, or 10.01%.

To determine if this is unusual, we need to consider the context or benchmark. If we define "unusual" as a probability less than 5% (0.05), then finding that all 8 randomly selected adults have health insurance would indeed be considered unusual.

However, it's important to note that what is considered unusual may vary depending on the specific context or criteria being used.

To find the probability that 8 adults selected at random from the town all have health insurance, we can use the binomial probability formula:

P(X = k) = n C k * p^k * (1 - p)^(n - k)

Where:
- n is the number of trials (8 in this case, since we are selecting 8 adults)
- k is the number of successful outcomes (all 8 adults having health insurance)
- p is the probability of a successful outcome on a single trial (76% or 0.76, since 76% of adults have health insurance)

Plugging in the values, we get:

P(X = 8) = 8 C 8 * 0.76^8 * (1 - 0.76)^(8 - 8)
= 1 * 0.76^8 * 0.24^0
= 0.76^8
≈ 0.2288

Therefore, the probability that 8 randomly selected adults all have health insurance in this town is approximately 0.2288 or 22.88%.

Now, to determine if it's unusual or not, we need to compare this probability to a certain threshold. Commonly, a probability below 5% is considered unusual. However, this threshold may vary depending on the context.

In this case, 0.2288 (or 22.88%) is well above the 5% threshold, so it would not be considered unusual to select 8 people at random and find that they all have health insurance. This suggests that having all 8 people with health insurance is relatively likely in this town given the provided probability.