Question

Three coins are tossed simultaneously. Consider the event E ‘three heads or three tails’, F ‘at least two heads' and G ‘at most two heads’. Of the pairs (E, F), (E, G) and (F, G), which are independent? Which are dependent?

Solution

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
$therefore space space$ E = {HHH, TTT}, F = {HHH, HHT, HTH, THH}
and G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
$therefore space$ $straight E intersection straight F space equals space left curly bracket HHH right curly bracket comma space straight E space intersection space straight G space equals space left curly bracket TTT right curly bracket comma space space straight F intersection straight G space equals space left curly bracket HHT comma space space HTH comma space THH right curly bracket$
$therefore space space space straight P left parenthesis straight E right parenthesis space equals space 2 over 8 space equals space 1 fourth comma space space straight P left parenthesis straight F right parenthesis space equals space 4 over 8 space equals 1 half comma space space straight P left parenthesis straight G right parenthesis space equals space 7 over 8$
and $straight P left parenthesis straight E intersection straight F right parenthesis space equals space 1 over 8 comma space space straight P left parenthesis straight E intersection straight G right parenthesis space equals space 1 over 8 comma space space straight P left parenthesis straight F intersection straight G right parenthesis space equals space 3 over 8$
Also, $straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F right parenthesis space equals space 1 fourth cross times 1 half space equals 1 over 8 comma space space straight P left parenthesis straight E right parenthesis. space space straight P left parenthesis straight G right parenthesis space equals space 1 fourth cross times 7 over 8 space equals 7 over 32$
and $straight P left parenthesis straight F right parenthesis. space straight P left parenthesis straight G right parenthesis space equals 1 half cross times 7 over 8 equals space 7 over 16$
$therefore space space space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F right parenthesis$
$straight P left parenthesis straight E space intersection space straight G right parenthesis thin space not equal to space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight G right parenthesis$
and $straight P left parenthesis straight F intersection straight G right parenthesis space not equal to space straight P left parenthesis straight F right parenthesis. space straight P left parenthesis straight G right parenthesis$
∴ the events (E and F) are independent, and the events (E and G) and (F and G) are dependent.

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Probability

Three coins are tossed simultaneously. Consider the event E ‘three heads or three tails’, F ‘at least two heads' and G ‘at most two heads’. Of the pairs (E, F), (E, G) and (F, G), which are independent? Which are dependent?

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