Independence Show the formula you use to check these a You r

Independence: Show the formula you use to check these!
(a) You roll a six-sided die. Are the following two events independent?

A = “the number is less than or equal to 4”,

B = “the number is odd”.

(b) You flip two coins. Are the following two events independent?

A = “you flipped repeats (HH or TT)”,

B = “you flipped at least one heads”.

If events A and B are independent, then

P(A|B) = P(A)
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Here, A is satisfied by {1,2,3,4}.

P(A) = 4/6 = 2/3

Meanwhile, B = {1,3,5}. Of these, two satisfy A. Thus,

P(A|B) = 2/3

Thus, A and B are independent. [ANSWER]

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There are 2*2 = 4 possibilities in tossing 2 coins. Hence,

P(A) = 2/4 = 1/2

Meanwhile, B = {HT, TH, HH}. Of these, just 1 is a repeat.

Thus,

P(A|B) = 1/3

As P(A) =/= P(A|B), THEY ARE NOT INDEPENDENT. [ANSWER]