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Example 12 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) & (F, G), which are independent? which are dependent? Three coins are tossed simultaneously S = {(H, H, H), (H, H, T), (T, H, H), (H, T, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)} Given 3 events as E : 3 head or 3 tails F : atleast two heads G : atmost two heads Finding probabilities of E, F and G Event E E : 3 head or 3 tails E = {HHH, TTT} P(E) = 2/8 = 1/4 Event F F : atleast two heads F = {HHH, HHT, HTH, THH} P(F) = 4/8 = 1/2 Event G G : atmost two heads G ={HHT, HTH, THH HTT, THT, TTH ,TTT } P(G) = 7/8 Now, let us find Probabilities of E ∩ F , F ∩ G , E ∩ G E and F E ∩ F = {HHH} So, P(E ∩ F) = 1/8 Now, P(E) . P(F) = 1/4 × 1/2 = 1/8 = P(E ∩ F) P(E ∩ F) = P(E).P(F) Thus, E & F are independent events F and G F ∩ G = {HHH, HTH, THH} So, P(F ∩ G) = 3/8 Now, P(F) . P(G) = 1/2 × 7/8 = 7/16 P (F ∩ G) ≠ P(F) . P(G) Thus, F & G are not independent events E and G E ∩ G = {TTT} So, P(E ∩ G) = 1/8 Now, P(E) . P(G) = 1/4 × 7/8 = 7/32 P (E ∩G) ≠ P (E). P(G) Thus, E & G are not independent events

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo