Question 1209733
.
60% of all violent felons in the prison system are repeat offenders. 
If 45 violent felons are randomly selected, find the probability that

a. Exactly 26 of them are repeat offenders.
b. At most 29 of them are repeat offenders.
c. At least 29 of them are repeat offenders.
d. Between 25 and 33 (including 25 and 33) of them are repeat offenders.
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        As it regularly happens with  @CPhill' solutions on  Binomial distribution,

        all calculations  (a),  (b),  (c)  and  (d)  in the post by  @CPhill are incorrect.


        Below is my correct solution.



<pre>
Here's how to solve this binomial probability problem:

* **n** (number of trials) = 45
* **p** (probability of success - repeat offender) = 0.60
* **q** (probability of failure - not a repeat offender) = 1 - p = 0.40

The binomial probability formula is: P(x) = (nCx) * p^x * q^(n-x)

Where nCx represents "n choose x" (the binomial coefficient).

**(a) Exactly 26 are repeat offenders:**

P(x = 26) = (45C26) * (0.60)^26 * (0.40)^19
P(x = 26) ≈ 0.1143

**(b) At most 29 are repeat offenders:**

This means 0 to 29 are repeat offenders. This is a cumulative probability; we need P(x ≤ 29).  
It's best to use a binomial cumulative distribution function (CDF) calculator or statistical software for this.

P(x ≤ 29) ≈ 0.7751

**(c) At least 29 are repeat offenders:**

This means 29 to 45 are repeat offenders. We can use the complement rule:

P(x ≥ 29) = 1 - P(x < 29) = 1 - P(x ≤ 28)

Use a binomial CDF calculator:
P(x ≥ 29) = 1 - 0.6728
P(x ≥ 29) ≈ 0.3272

**(d) Between 25 and 33 (inclusive):**

This means 25, 26, 27, 28, 29, 30, 31, 32, and 33 are repeat offenders. We can use the CDF:

P(25 ≤ x ≤ 33) = P(x ≤ 33) - P(x ≤ 24)

Use a binomial CDF calculator:
P(25 ≤ x ≤ 33) = 0.9784 - 0.2223
P(25 ≤ x ≤ 33) ≈ 0.7561

**Summary of Answers:**

* (a) P(x = 26) ≈ 0.1143
* (b) P(x ≤ 29) ≈ 0.7751
* (c) P(x ≥ 29) ≈ 0.3272
* (d) P(25 ≤ x ≤ 33) ≈ 0.7561
</pre>

Solved CORRECTLY.


Practically all calculations by @CPhill to this kind of problems on Binomial distribution are INCORRECT.


To check your/my calculations, &nbsp;use online calculator 

https://stattrek.com/online-calculator/binomial



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;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.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



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


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


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


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


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


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.