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發問:
Q1) There are 6 balls: one red and five white. John and Mary draw ball one by one without replacement. Which party gets the red ball first, he or she will become the winner of the game. John draws first, then which party has higher probability to win that game?Q2) In the figure, PA and PB touch the circle at A... 顯示更多 Q1) There are 6 balls: one red and five white. John and Mary draw ball one by one without replacement. Which party gets the red ball first, he or she will become the winner of the game. John draws first, then which party has higher probability to win that game? Q2) In the figure, PA and PB touch the circle at A and B respectively.
最佳解答:
1. J : John gets the red ball j : John gets the white ball M : Mary gets the red ball m : Mary gets the white ball P(John wins) = P(J or jmJ or jmjmJ) = P(J) + P(jmJ) + P(jmjmJ) = (1/6) + (5/6) x (4/5) x (1/4) + (5/6) x (4/5) x (3/4) x (2/3) x (1/2) = (1/6) + (1/6) + (1/6) = 1/2 P(Mary wins) = 1 - P(John wins) = 1 - (1/2) = 1/2 John and Mary have equal probabilityto win the game. ===== 2. Let O be the centre of the circle. Join OA and OB. Angle at circumference ∠ACB = 55° Angle at centre is twice the angle at circumference subtending the same arc. Angle at centre ∠AOB= 55° x 2 = 110° Denote θ as the reflex angle of ∠AOB. θ+ ∠AOB = 360° (∠s at a pt.) θ + 110° = 360° θ = 250° In quadrilateral ACBO : θ + ∠OAC + ∠OAB + ∠ACB = 360° (sum of int. ∠sof quad.) 250° + ∠OAC + ∠OAB + 55° = 360° ∠OAC + ∠OAB = 55° The radius ⊥ the tangent at the point of contact. ∠PAO = ∠PBO = 90° ∠a + ∠b = (∠PAO + ∠OAC) + (∠PBO + ∠OBC) = (90° + ∠OAC) + (90° + ∠OBC) = 180° + (∠OAC + ∠OBC) = 180° + 55° = 235°
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F.5 MATHS (2Qs)發問:
Q1) There are 6 balls: one red and five white. John and Mary draw ball one by one without replacement. Which party gets the red ball first, he or she will become the winner of the game. John draws first, then which party has higher probability to win that game?Q2) In the figure, PA and PB touch the circle at A... 顯示更多 Q1) There are 6 balls: one red and five white. John and Mary draw ball one by one without replacement. Which party gets the red ball first, he or she will become the winner of the game. John draws first, then which party has higher probability to win that game? Q2) In the figure, PA and PB touch the circle at A and B respectively.
最佳解答:
1. J : John gets the red ball j : John gets the white ball M : Mary gets the red ball m : Mary gets the white ball P(John wins) = P(J or jmJ or jmjmJ) = P(J) + P(jmJ) + P(jmjmJ) = (1/6) + (5/6) x (4/5) x (1/4) + (5/6) x (4/5) x (3/4) x (2/3) x (1/2) = (1/6) + (1/6) + (1/6) = 1/2 P(Mary wins) = 1 - P(John wins) = 1 - (1/2) = 1/2 John and Mary have equal probabilityto win the game. ===== 2. Let O be the centre of the circle. Join OA and OB. Angle at circumference ∠ACB = 55° Angle at centre is twice the angle at circumference subtending the same arc. Angle at centre ∠AOB= 55° x 2 = 110° Denote θ as the reflex angle of ∠AOB. θ+ ∠AOB = 360° (∠s at a pt.) θ + 110° = 360° θ = 250° In quadrilateral ACBO : θ + ∠OAC + ∠OAB + ∠ACB = 360° (sum of int. ∠sof quad.) 250° + ∠OAC + ∠OAB + 55° = 360° ∠OAC + ∠OAB = 55° The radius ⊥ the tangent at the point of contact. ∠PAO = ∠PBO = 90° ∠a + ∠b = (∠PAO + ∠OAC) + (∠PBO + ∠OBC) = (90° + ∠OAC) + (90° + ∠OBC) = 180° + (∠OAC + ∠OBC) = 180° + 55° = 235°
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