-- Jan 30 In-Class Exercise Thread
Sample Space S = {HH, HT, TH, TT}
Pr{X} = First Head = 1/2 (HH, HT)
Pr{Y} = Second Head = 1/2 (HH, TH)
Pr{Z} = Different = 1/2 (HT, TH)
PAIRWISE INDEPENDENCE
Pr{XnY} = Pr{X} * Pr{Y} = 1/2 * 1/2 = 1/4
There is a 1/4 chance the outcome is HH.
Pr{XnZ} = Pr{X} * Pr{Z} = 1/2 * 1/2 = 1/4.
There is a 1/4 chance the outcome is HT.
Pr{YnZ} = Pr{Y} * Pr{Z} = 1/2 * 1/2 = 1/4.
There is a 1/4 chance the outcome is TH.
NOT MUTUALLY INDEPENDENT
Pr{XnYnZ} does not equal Pr{X} * Pr{Y} * Pr{Z} = 1/2 * 1/2 * 1/2 = 1/8
There is a 0 chance the first toss is heads, the second toss is heads, and the two flips are different.
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Edited: 2019-01-30)
Sample Space S = {HH, HT, TH, TT}
Pr{X} = First Head = 1/2 (HH, HT)
Pr{Y} = Second Head = 1/2 (HH, TH)
Pr{Z} = Different = 1/2 (HT, TH)
PAIRWISE INDEPENDENCE
Pr{XnY} = Pr{X} * Pr{Y} = 1/2 * 1/2 = 1/4
There is a 1/4 chance the outcome is HH.
Pr{XnZ} = Pr{X} * Pr{Z} = 1/2 * 1/2 = 1/4.
There is a 1/4 chance the outcome is HT.
Pr{YnZ} = Pr{Y} * Pr{Z} = 1/2 * 1/2 = 1/4.
There is a 1/4 chance the outcome is TH.
NOT MUTUALLY INDEPENDENT
Pr{XnYnZ} does not equal Pr{X} * Pr{Y} * Pr{Z} = 1/2 * 1/2 * 1/2 = 1/8
There is a 0 chance the first toss is heads, the second toss is heads, and the two flips are different.