A digital scale that provides weights to the nearest gram is used. a. What is the sample space for this experiment? Let A denote the event that a weight exceeds 11 grams, let B denote the event that a weight is less than or equal to 15 grams, and let C denote the event that a weight is greater than or equal to 8 grams and less than 12 grams. Describe the following events. b. A ∪ B c. A ∩ B d. A' e. A ∪ B ∪ C f. (A ∪ C)' g. A ∩ B ∩ C h. B' ∩ C i. A ∪ (B ∩ C)

c) A ∩ B ="a weight exceeds 11 grams AND is less than or equal to 15 grams" and that is represent by [tex] 11 < X \leq 15 [/tex] who is the same as [tex] 12 \leq X \leq 15[/tex] .

d) A' = "a weight NOT exceeds 11 grams" [tex] X \leq 11[/tex] that's the complement of the event A

e) A ∪ B ∪ C = "represent all the possib;e values for the sample space or S"

f) (A ∪ C)'="for this case (AUC) represent the weigths that exceeds 11 gr OR are between 8 and less than 11, so on this case values [tex] X \geq 8[/tex], so then the complement (AUC)' woudl be all the values [tex] X <8[/tex]"

g) A ∩ B ∩ C =[tex]\emptyset[/tex] since we don't have a common interval for the 3 events at the same time

h) B' ∩ C = The complement of B are the [tex] X>15[/tex] and for C we have values [tex] 8 \leq X <12[/tex] and the intersection between these two events is the [tex]\emptyset[/tex].

i) A ∪ (B ∩ C) = For this case (B ∩ C) represent the values between [tex] 8\leq X <12[/tex] and if we do the union A ∪ (B ∩ C) we got [tex] X \geq 8[/tex]

Step-by-step explanation:

For this case we have defined the following events, assuming that X represent the weight:

A= "a weight exceeds 11 grams" [tex]X>11[/tex]

B= " a weight is less than or equal to 15 grams" [tex] X \leq 15[/tex]

C= "a weight is greater than or equal to 8 grams and less than 12 grams" [tex] 8 \leq X < 12[/tex]

Part a

The sample space is given by:

[tex] S= [x \in x \geq 0][/tex]

Because the weigth can't be negative.

Part b

AUB = "a weight exceeds 11 grams OR is less than or equal to 15 grams" and that is represent by all the sample space S.

Part c

A ∩ B ="a weight exceeds 11 grams AND is less than or equal to 15 grams" and that is represent by [tex] 11 < X \leq 15 [/tex] who is the same as [tex] 12 \leq X \leq 15[/tex] .

Part d

A' = "a weight NOT exceeds 11 grams" [tex] X \leq 11[/tex] that's the complement of the event A

Part e

A ∪ B ∪ C = "represent all the possib;e values for the sample space or S"

Part f

(A ∪ C)'="for this case (AUC) represent the weigths that exceeds 11 gr OR are between 8 and less than 11, so on this case values [tex] X \geq 8[/tex], so then the complement (AUC)' woudl be all the values [tex] X <8[/tex]"

Part g

A ∩ B ∩ C =[tex]\emptyset[/tex] since we don't have a common interval for the 3 events at the same time

Part h

B' ∩ C = The complement of B are the values in[tex] X>15[/tex] and for C we have values [tex] 8 \leq X <12[/tex] and the intersection between these two events is the [tex]\emptyset[/tex].

Part i

A ∪ (B ∩ C) = For this case (B ∩ C) represent the values between [tex] 8\leq X <12[/tex] and if we do the union A ∪ (B ∩ C) we got [tex] X \geq 8[/tex]