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the event that the member takes part in croquet. Given that P(B) = 0.45: a find x b find y. = A or B 'union' AUB a) P (B) = 0.45 0.45-0.2-0.1 = 0.15 = x b) 0.45 + 0.35 +0.05= 0.85 5/6 C b) 1) 25/30 ii) 20/30 = 2/3 CC 0.35 0.1 0.2 y 0.05 (1 mark) (2 marks) 10.85 = 0.15 = y ∞ not A 'complement A' Mutually exclusive ⇒ no outcomes in common P(AUB) P(A) + P(B) Independant → don't affect each other P(An B) = P(A) X P(B) The Venn diagram shows the number of students in a particular class who watch any of three popular TV programmes. a Find the probability that a student chosen at random watches B or C or both. b Determine whether watching A and watching B are statistically independent. a) 4+ 5 + 10 + 7 = 26 probability = = 26/30 i P(S and T) a) PS and T) = Pls and T) = 13/15 ii P(neither S nor T). = P(S) X P(T) 7 Sand Tare two events such that P(S) = 0.3. P(7) = 0.4 and P(S but not 7) = 0.18. a Show that S and T are independent. b Find: E/P 9 The Venn diagram shows the probabilities of members of a social club taking part in charitable activities. A represents taking part in an archery competition. R represents taking part in a raffle. F represents taking part in a fun run. The probability that a member takes part in the archery competition or the raffle is 0.6. a Find the value of x and the value of y. b Show that events R and Fare not independent. a) P(A or R) = P(A). 0.6 = 0.2 0.15 = 3√ 3 P(S) - P(s not T) = 0.3-0.18 = 0.12 P(S) x P(T) = 0.3x0.4 = 0.12 ⇒ events are independant. b) Pls and T) = 0.12 ✓ P(neithe Snor T) = 1-0.4 -0.18 = 0.42 ✓ + P(B) + 0.25 + X b) independant P(A and B) = P(A) × P(B) P(A) = 7/30 P(B)=19/30 P(A and B) = 30 7 19 X 133 30 30 900 #4 30 :: A and B are not independant. b) P(Rand F) = P(R) × P(F) P(R)= 0.4 P(F) = 0.45 0.4 x 0.45 = 0.18 B addition rule P(AUB) P(AUB) = P(A) + P(B) - P(ANB) 5 10 0.2 0.1 OO P(ANB)=O 0.25 (2 marks) (3 marks) 1-0.2-0.25 -0.15-0.1 = 0.3 y=0.3✓ P(Rand F) = 0.15 0.15 :: events are not independant. Tree Diagrams → events happening in succesion 3 The probability that Charlie takes the bus to school is 0.4. If he doesn't take the bus, he walks. The probability that Charlie is late to school if he takes the bus is 0.2. The probability he is late to school if he walks is 0.3. a Draw a tree diagram to represent this information. b Find the probability that Charlie is late to school. 0.2 late a) 0.4 bus walk 0.6 H 1/3 12/3 0.8 5 A biased coin is tossed three times and it is recorded whether it falls heads or tails. P(heads) = a Draw a tree diagram to represent this experiment. b Find the probability that the coin lands on heads all three times. e Find the probability that the coin lands on heads only once The whole experiment is repeated for a second trial. d Find the probability of obtaining either 3 heads or 3 tails in both trials. 03 late not late 0.7 not late 113 H. 123 4½ T 2/3 T43 Узн узн Узн 12/m 31 6) bus-late = 0.4x 0.2 = 0.08 walk-late = 0.6 x 0.3= 0.18 P(late) = 0.26 T (3 marks) (1 mark) (2 marks) (3 marks) 6) 1/3 x 1/3 X ¹1/13 = ½¼/27 ✓ x in both trials 4 C) 1/3 x ²/3 x ²3 x 3 = 2√ d) P(HHH or TTT) = 1/3 x 1/3 = 1/9 v 13x13x1/3 + 23x²3x²/3 = 1/3