In a large survey, it was discovered that 1 out of every 5 adults visits Disney World every year. If 30 adults are randomly selected, what is the probability that exactly 7 of them will visit Disney World this year?
To find the probability, we need to use the binomial probability formula. The formula is given by:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of getting exactly x successes,
C(n, x) is the number of combinations of n items taken x at a time (also known as the binomial coefficient),
p is the probability of success on a single trial,
n is the number of trials.
In this case, the probability of success (visiting Disney World) is 1/5, since 1 out of every 5 adults visit Disney World. So, p = 1/5.
The number of trials is 30, as 30 adults are randomly selected. So, n = 30.
We want to find the probability of exactly 7 adults visiting Disney World, so x = 7.
Now, let's calculate the probability using the formula:
P(7) = C(30, 7) * (1/5)^7 * (4/5)^(30 - 7)
To calculate C(30, 7), we can use the formula for combination:
C(n, x) = n! / (x!(n - x)!)
C(30, 7) = 30! / (7!(30 - 7)!)
By calculating the above expression, we find that C(30, 7) equals 203,580.
Now we can substitute the values into the formula:
P(7) = 203,580 * (1/5)^7 * (4/5)^(30 - 7)
Calculating this expression will give us the probability that exactly 7 of the 30 adults will visit Disney World this year.