At a particular hospital,records show that each day, on average,80% of people keep their appointment at the outpatient's clinic. i)Find the probability that in a random sample of 10 patients, more than 7 keep their appointment.
prob(keep) = 80/100 = 4/5
prob(not keep) = 1/5
prob(10 patients, more than 7 keep their appointment.)
= prob(8 will keep) + prob(9 will keep) + prob(10 will keep)
= C(10,8) (4/5)^8 (1/5)^2 + C(10,9) (4/5)^9 (1/5) + C(10,10) (4/5)^10 (1/5)^0
= 45(.16777216)(.04) + ...... + 1(.107374182)(1)
= ....
let me know what you get
Oh, probability and statistics. They always keep me on my toes!
To find the probability that more than 7 people out of 10 keep their appointment at the outpatient's clinic, we need to do a little math.
So, let's break it down step by step. The probability that any given patient keeps their appointment is 80%, which we can express as 0.8.
Now, in order to find the probability of more than 7 patients out of 10 keeping their appointment, we need to calculate the probabilities for 8, 9, and 10 patients separately, and then add them all up.
Let's get crunching those numbers, shall we?
The probability of exactly 8 patients out of 10 keeping their appointment can be calculated using the binomial distribution formula. It's 10 (the number of patients) choose 8 (the number of desired successes) multiplied by 0.8 (probability of success) to the power of 8, multiplied by 0.2 (probability of failure) to the power of 2 (the remaining failures, which is 10 - 8).
Now, repeat this process for 9 patients and 10 patients:
- For 9 patients out of 10: 10 choose 9 multiplied by 0.8 to the power of 9, multiplied by 0.2 to the power of 1.
- For 10 patients out of 10: 10 choose 10 multiplied by 0.8 to the power of 10, multiplied by 0.2 to the power of 0 (which is always equal to 1).
Once you've calculated these probabilities, add them all together to get the final probability. That'll give you the answer you're looking for!
And remember, even though math can be a little daunting at times, you got this! Good luck!
To find the probability that more than 7 patients keep their appointment, we need to calculate the probability of having 8, 9, or 10 patients keeping their appointment.
Let's use the binomial probability formula to calculate the probability:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) represents the probability of having k patients keeping their appointment
- C(n, k) represents the number of ways to choose k successes from n trials
- p represents the probability of success (keeping the appointment)
- n represents the number of trials (sample size)
Given that p = 0.8 (80% of people keep their appointment) and n = 10, we can now calculate the probability.
P(X > 7) = P(X = 8) + P(X = 9) + P(X = 10)
P(X = 8) = C(10, 8) * (0.8)^8 * (1 - 0.8)^(10 - 8)
P(X = 9) = C(10, 9) * (0.8)^9 * (1 - 0.8)^(10 - 9)
P(X = 10) = C(10, 10) * (0.8)^10 * (1 - 0.8)^(10 - 10)
Now, let's calculate each term:
P(X = 8) = 10C8 * (0.8)^8 * (0.2)^2
= 45 * 0.16777216 * 0.04
= 0.301989888
P(X = 9) = 10C9 * (0.8)^9 * (0.2)^1
= 10 * 0.134217728 * 0.2
= 0.026843546
P(X = 10) = 10C10 * (0.8)^10 * (0.2)^0
= 1 * 0.1073741824 * 1
= 0.1073741824
Now, calculate the overall probability:
P(X > 7) = P(X = 8) + P(X = 9) + P(X = 10)
= 0.301989888 + 0.026843546 + 0.1073741824
= 0.4362076164
Therefore, the probability that in a random sample of 10 patients, more than 7 keep their appointment is approximately 0.436 or 43.6%.
To solve this problem, you can use the binomial probability formula. The formula for the probability of getting exactly k successes in n trials with a probability of success p is:
P(X = k) = C(n, k) * (p^k) * ((1 - p)^(n - k))
Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success,
- k is the number of successes, and
- n is the number of trials.
Let's apply this formula to find the probability that more than 7 patients keep their appointment.
Step 1: Calculate the probability of success (p):
The problem states that, on average, 80% of people keep their appointment. Therefore, p = 0.8.
Step 2: Determine the number of trials (n):
We have a random sample of 10 patients. Therefore, n = 10.
Step 3: Calculate the probability of getting more than 7 successes:
To find the probability of more than 7 successes, we need to calculate the probability of getting 8, 9, or 10 successes. We can add up the probabilities for each case individually.
P(X > 7) = P(X = 8) + P(X = 9) + P(X = 10)
P(X > 7) = C(10, 8) * (0.8^8) * ((1 - 0.8)^(10 - 8))
+ C(10, 9) * (0.8^9) * ((1 - 0.8)^(10 - 9))
+ C(10, 10) * (0.8^10) * ((1 - 0.8)^(10 - 10))
You can evaluate this expression to find the final answer.