if 20%of the people in school use the emergency room at the hospital in one year. find the probabilities for a sample of 8 people
To find the probabilities for a sample of 8 people, we need to use the concept of binomial distribution. In this case, we have two possible outcomes for each person: either they use the emergency room (success) or they don't use the emergency room (failure).
The probability of success (p) is given as 20%, which can also be expressed as 0.2. The probability of failure (q) is calculated as 1 - p, which in this case is 1 - 0.2 = 0.8.
Using the binomial distribution formula, the probability (P) of obtaining exactly x successes in a sample of n individuals is calculated as:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- nCx refers to the number of combinations of n items taken x at a time, and it is calculated as n! / (x! * (n-x)!), using the factorial function (!) to represent the product of all positive integers up to that number.
- p^x refers to the probability of x successes occurring.
- q^(n-x) refers to the probability of (n-x) failures occurring.
Now, let's find the probabilities for a sample of 8 people using the given information:
P(x=0) = (8C0) * (0.2^0) * (0.8^8)
P(x=1) = (8C1) * (0.2^1) * (0.8^7)
P(x=2) = (8C2) * (0.2^2) * (0.8^6)
P(x=3) = (8C3) * (0.2^3) * (0.8^5)
P(x=4) = (8C4) * (0.2^4) * (0.8^4)
P(x=5) = (8C5) * (0.2^5) * (0.8^3)
P(x=6) = (8C6) * (0.2^6) * (0.8^2)
P(x=7) = (8C7) * (0.2^7) * (0.8^1)
P(x=8) = (8C8) * (0.2^8) * (0.8^0)
To calculate these probabilities, substitute the values into the formula and simplify the calculations.