In one town, 61% of adults have health insurance. What is the probability that 6 adults selected at random from the town all have health insurance?
To find the probability that 6 adults selected at random from the town all have health insurance, you can use the binomial probability formula. Here's how you can calculate it:
Step 1: Determine the probability of success (having health insurance).
Given that 61% of adults have health insurance, the probability of an adult selected at random having health insurance is 0.61.
Step 2: Determine the number of trials (adults selected).
In this case, we are selecting 6 adults randomly.
Step 3: Plug in the values into the binomial probability formula.
The formula for the probability of x successes in n trials can be represented as:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) represents the probability of x successes,
n represents the number of trials,
p represents the probability of success for each trial,
(nCx) represents the binomial coefficient, which is the number of ways to choose x successes from n trials.
In our case:
n = 6 (number of adults selected)
x = 6 (number of adults with health insurance)
p = 0.61 (probability of an adult having health insurance)
Substituting these values into the formula, we get:
P(6) = (6C6) * 0.61^6 * (1-0.61)^(6-6)
Step 4: Calculate the binomial coefficient.
The binomial coefficient (nCx) represents the number of ways to choose x successes from n trials. In this case, when selecting 6 adults out of 6, there is only one way this can happen. So, we have (6C6) = 1.
Step 5: Calculate the probability.
Now, plug in the values and calculate:
P(6) = 1 * 0.61^6 * (1-0.61)^(6-6)
P(6) = 0.61^6
Using a calculator, you can find the value of 0.61^6, which is approximately 0.0408946.
Therefore, the probability that 6 adults selected at random from the town all have health insurance is approximately 0.0408946 or 4.09%.