F.5 MATHS (2Qs)

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.


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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°