If 20% of the people in a community use the emergency room at a hospital in one year, find the probabilities for a sample of 10 people,

a.At the most three use the emergency room.

b.Exactly three use the emergency room

c.At least five use the emergency room.

To find the probabilities for a sample of 10 people using the emergency room, we need to use the binomial probability formula. The binomial probability formula is:

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

Where:
P(X=k) is the probability of getting exactly k successes (people who use the emergency room)
n is the total number of trials (10 people in this case)
k is the number of successes (people who use the emergency room)
p is the probability of success (20% or 0.20 in this case)
(1-p) is the probability of failure (80% or 0.80 in this case)
nCk is the binomial coefficient, which represents the number of ways to choose k successes from n trials and can be calculated using the formula: nCk = n! / (k!(n-k)!)

Now let's calculate the probabilities for each scenario step-by-step:

a. At the most three use the emergency room:
Since we want to find the probability of at most three people using the emergency room, we need to calculate the probabilities for 0, 1, 2, and 3 people using the formula mentioned above.

P(X <= 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

P(X = 0) = (10C0) * (0.20)^0 * (1-0.20)^(10-0)
P(X = 1) = (10C1) * (0.20)^1 * (1-0.20)^(10-1)
P(X = 2) = (10C2) * (0.20)^2 * (1-0.20)^(10-2)
P(X = 3) = (10C3) * (0.20)^3 * (1-0.20)^(10-3)

b. Exactly three use the emergency room:
For finding the probability of exactly three people using the emergency room, we simply calculate P(X = 3) using the formula mentioned above.

P(X = 3) = (10C3) * (0.20)^3 * (1-0.20)^(10-3)

c. At least five use the emergency room:
To find the probability of at least five people using the emergency room, we calculate the probabilities for 5, 6, 7, 8, 9, and 10 people, and then add them up.

P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

P(X = 5) = (10C5) * (0.20)^5 * (1-0.20)^(10-5)
P(X = 6) = (10C6) * (0.20)^6 * (1-0.20)^(10-6)
P(X = 7) = (10C7) * (0.20)^7 * (1-0.20)^(10-7)
P(X = 8) = (10C8) * (0.20)^8 * (1-0.20)^(10-8)
P(X = 9) = (10C9) * (0.20)^9 * (1-0.20)^(10-9)
P(X = 10) = (10C10) * (0.20)^10 * (1-0.20)^(10-10)

Let's calculate these probabilities step-by-step.

To find the probabilities for a sample of 10 people using the emergency room, we can use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes (success or failure) and each trial is independent.

Let's define the variables:
n = number of trials (number of people in the sample) = 10
p = probability of success (proportion of people using the emergency room) = 0.20

Now, let's apply the binomial distribution formula to solve each part of the problem:

a. To find the probability that at most three people use the emergency room, we need to sum up the probabilities of having 0, 1, 2, or 3 people using the emergency room.

P(at most three) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

To calculate individual probabilities, we can use the binomial probability formula:

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

where (n C k) represents the combination formula.

Using this formula for each case, we get:

P(X=0) = (10 C 0) * (0.20^0) * (0.80^10)

P(X=1) = (10 C 1) * (0.20^1) * (0.80^9)

P(X=2) = (10 C 2) * (0.20^2) * (0.80^8)

P(X=3) = (10 C 3) * (0.20^3) * (0.80^7)

b. To find the probability that exactly three people use the emergency room, we need to calculate:

P(exactly three) = P(X=3)

c. To find the probability that at least five people use the emergency room, we can sum up the probabilities of having 5, 6, 7, 8, 9, or 10 people using the emergency room.

P(at least five) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

Now, you can substitute the values into the binomial probability formula to calculate the probabilities for each part of the problem.

If 20% of the people in a community use the emergency room at a hospital in one year, find the probabilities for a sample of 10 people,

a.At the most three use the emergency room.

b.Exactly three use the emergency room

c.At least five use the emergency room.