SOLUTION: If 35% of the people in a community use the emergency room at a hospital one year. Find these probabilities for a sample of 12 people A) exactly 4 used the emergency room B) At mo

Question 1179602: If 35% of the people in a community use the emergency room at a hospital one year. Find these probabilities for a sample of 12 people A) exactly 4 used the emergency room B) At most 4 used the emergency room C) At least 10 used the emergency room
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
This problem can be solved using the binomial probability distribution.
**Given:**
* Probability of success (using the emergency room): p = 0.35
* Probability of failure (not using the emergency room): q = 1 - p = 1 - 0.35 = 0.65
* Sample size: n = 12
**Binomial Probability Formula:**
P(X = k) = (nCk) * p^k * q^(n-k)
Where:
* nCk = n! / (k! * (n-k)!) (the number of combinations of n items taken k at a time)
* k = number of successes
* n = sample size
**A) Exactly 4 used the emergency room (P(X = 4))**
* P(X = 4) = (12C4) * (0.35)^4 * (0.65)^8
* 12C4 = 12! / (4! * 8!) = 495
* P(X = 4) = 495 * (0.01500625) * (0.031864089)
* P(X = 4) ≈ 0.2366
**B) At most 4 used the emergency room (P(X ≤ 4))**
This means we need to find P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
* P(X = 0) = (12C0) * (0.35)^0 * (0.65)^12 ≈ 0.0037
* P(X = 1) = (12C1) * (0.35)^1 * (0.65)^11 ≈ 0.0261
* P(X = 2) = (12C2) * (0.35)^2 * (0.65)^10 ≈ 0.0844
* P(X = 3) = (12C3) * (0.35)^3 * (0.65)^9 ≈ 0.1744
* P(X = 4) ≈ 0.2366 (calculated in part A)
* P(X ≤ 4) ≈ 0.0037 + 0.0261 + 0.0844 + 0.1744 + 0.2366 ≈ 0.5252
**C) At least 10 used the emergency room (P(X ≥ 10))**
This means we need to find P(X = 10) + P(X = 11) + P(X = 12).
* P(X = 10) = (12C10) * (0.35)^10 * (0.65)^2 ≈ 0.00018
* P(X = 11) = (12C11) * (0.35)^11 * (0.65)^1 ≈ 0.000013
* P(X = 12) = (12C12) * (0.35)^12 * (0.65)^0 ≈ 0.0000003
* P(X ≥ 10) ≈ 0.00018 + 0.000013 + 0.0000003 ≈ 0.0001933
**Summary:**
* A) P(X = 4) ≈ 0.2366
* B) P(X ≤ 4) ≈ 0.5252
* C) P(X ≥ 10) ≈ 0.0001933

Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

All calculation in the post by @CPhill related to parts (b) and (c) of this problem are INCORRECT.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

                Regarding the post by @CPhill . . .

Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.

The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.

                It has no feeling of shame - it is shameless.

This time, again, it made an error.

Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.

Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.

All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.

Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.

And the last my comment.

When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.

Without it, their reliability is ZERO and their creadability is ZERO, too.

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