Question 1209735
.
69% of all Americans are home owners. If 37 Americans are randomly selected, find the probability that
a. Exactly 27 of them are are home owners.
b. At most 27 of them are are home owners.
c. At least 23 of them are home owners.
d. Between 23 and 28 (including 23 and 28) of them are home owners.
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        Calculations in the post by @CPhill  ALL  are  INCORRECT, 
        including each and every part  (a),  (b),  (c)  and  (d).


        Below are my correct answers.



<pre>
(a)  P(n=37; k = 27, p = 0.69) = {{{C[37]^27*0.69^27*(1-0.69)^10}}} = {{{348330136*0.69^27*0.31^10}}} = 0.1272 (rounded).


(b)  P(n=37; k <=27, p=0.69) = 0.75415  (rounded).


(c)  P(n=37; k >= 23; p=0.69) = use cumulative formula = 1 - P(k <= 22) = 1 - 0.1412 = 0.8588    (rounded).


(d)  P(n=37; 23 <= k <= 27) = P(k <= 27) - P(k <= 22) = use cumulative probabilities for k= 27 and k= 22   ==>

                            = 0.75415 - 0.1412 = 0.61295.
</pre>

Solved.


All these calculations can be EASILY done using an online free of charge calculator for binomial probabilities
https://stattrek.com/online-calculator/binomial/



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But if you want your head to split in half, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;     then follow @CPhill's instructions.



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I am chocked by seeing this gibberish, which @CPhill produced in his post.


If it is &nbsp;" Artificial Intelligence " - then &nbsp;Lord please save us from such helpers.



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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.



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Until now, &nbsp;I did not check the posts by @CPhill for probability, 
because I thought that they were made professionally.


But now I see that this attitude of mine was wrong,
because too many of his solutions/calculations are made extremely inaccurately,
so all these posts require careful checking.