Respuesta :
Answer:
[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]
The correct answer would be:
NO
Step-by-step explanation:
For this case we have the following dataset given
Hours   Number of students (f)
_______________________________
  4           15
  5           11
  6           19
  7            6
  8            9
  9           16
  10           2
______________________________
Total          78
For this case we have defined the following events:
A = event the student took at most 9 hours
B = event the student took at least 9 hours
And we can find the empirical probability for both elements like this:
[tex] P(A) = \frac{78-2}{78}= \frac{76}{78}[/tex]
[tex] P(B) = \frac{16+2}{78}= \frac{18}{78}[/tex]
And for this case we want to see if A and B are disjoint
From definition two events X and Y are disjoint if the two sets not have a common elements, and we satisfy that:
[tex] P(X \cap Y) =0[/tex]
So this case the intersection for the events A and B is X=9, because at most 9 means [tex] X \leq 9[/tex] and at least 9 means [tex] X \geq 9[/tex] and the intersection between [tex] X \leq 9[/tex] Â and [tex] X \geq 9[/tex] Â is X=9
So then the probability:
[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]
So then we can conclude that the two events not are disjoint
The correct answer would be:
NO
No, the events A and B are not disjoint.
If two events have no outcomes in common, then they are called disjoint.
We have data of the number of hours sixth grade students took to complete a research project as:
For this case we have the following dataset given
Hours   Number of students (f)
 4           15
5 Â Â Â Â Â Â Â Â Â Â 11
6 Â Â Â Â Â Â Â Â Â Â 19
7 Â Â Â Â Â Â Â Â Â Â Â 6
8 Â Â Â Â Â Â Â Â Â Â Â 9
9 Â Â Â Â Â Â Â Â Â Â 16
10 Â Â Â Â Â Â Â Â Â Â 2
Total          78
Two events are:
A = event the student took at most 9 hours
B = event the student took at least 9 hours
Now, the number of students who took at most 9 hours
= 78 - 2
= 76
So, [tex]P(A)=\frac{76}{78}[/tex]
The number of students who took at least 9 hours
=16 +2
=18
So, [tex]P(B)=\frac{16}{78}[/tex]
Number of students who read exactly 9 hours
P(A n B)[tex]=\frac{16}{78}[/tex][tex]\neq 0[/tex]
Therefore the events A and B disjoint are not disjoint.
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