PeluBoy wrote:

eto na lang, mas appropriate hi hi hi

In Probability Theory, let

A = the event that the numbers 1 to 7 all come up in a single local Ultra Lotto draw

B = the event that Bum Aquino wins in the 2019 senatorial elections

Are the events A and B independent? Provide proof whether yes or no.

Answer:

We start with stating the probability of event A, denoted by P(A). Event A is given as the numbers 1 to 7 coming up in a single Ultra Lotto draw, which will be impossible since there are only 6 numbers drawn per 6/58 draw. Hence,

P(A) = P(impossible event) = 0

Also, A is an empty set being an impossible event that contains no possible outcomes.

Now, let us assign the probability of B, P(B) with the constant x, such that 0 < x < 1...but were it not for this exercise, I'd really rather put the probability that Bum Aquino wins in the 2019 elections at simply 0 as well hihihihih

P(B) = x

Now, by Probability Theory definition, two events are independent "iff" the occurrence of one does not affect the probability of occurrence of the other. Mathematically, two events A and B are independent if and only if their joint probability equals the product of their probabilities:

P(A ∩ B) = P(A)P(B)

The equation above is what we need to prove, to show A and B are independent events.

Now, since A is an empty set, A = Ø

P(Ø ∩ B) = 0 * x

P(Ø) = 0 * x , since Ø ∩ B = Ø

0 = 0

We have shown that indeed P(A ∩ B) = P(A)P(B)

∴ A and B are independent

∎