The statement that correctly descrives the likelilhood of an event based on the given numbered cube is: Option A: It is likely that a number less than 6 is occured.
Suppose that for a considered sample space, we consider an event E
Then, E is likely to occur if:
[tex]P(E) \geq P(E')[/tex]
where [tex]E_i '[/tex] is complementary event of E (ie, E' occurs whenever E doesn't occur and vice versa)
If so, then we say that E is likely to occur as the outcome of the considered random experiment, else unlikely.
Since we have:
[tex]P(E) = 1- P(E')[/tex] due to E and E' being complement of each other, therefore, we get:
[tex]P(E) \geq 1 - P(E)\\P(E) \geq 0.5[/tex]
Thus, event E is said to be likely if its probability is greater or equal to 0.5, else not.
It is given that the cube is 6 faced (assuming all equally likely), numbered from 1 to 6.
Evaluating likelihood of each event:
P(Getting a number < 6 on that cube) = [tex]\dfrac{5}{6}[/tex] (since a number out of 6 numbers can occur in 6 ways, but it being < 6 can be done in 5 ways).
Thus, we have:
P(Getting a number < 6 on that cube) =5/6 > 0.5
So it is likely to get a number < 6 on the considered cube. So this statement correctly describes the likelihood of its considered event.
Ways to get a number on the considered cube = 6
Ways to get a number < 4 on the considered cube = 3
Thus, we get:
P(Getting a number < 4 on that cube) = 3/6 = 0.5
Thus, it is likely to get a number < 4 on that cube, so this option is wrong.
Yes it is certain, but we're considering only those options which correctly describe likelihood instead of certainity.
So, this option is not considerable.
Ways to get a number on the considered cube = 6
Ways to get a number > 1 on the considered cube = 5
Thus, we get:
P(Getting a number > 1 on that cube) = 5/6 > 0.5
Thus, it is likely to get a number > 1 on that cube, so this option is wrong.
Thus, the statement that correctly descrives the likelilhood of an event based on the given numbered cube is: Option A: It is likely that a number less than 6 is occured.
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