Ian Roblin Maths Resources

In the FA Cup Preliminary round, there are
n teams from both North & South. Find the probability of n all North v South ties.

Solution :
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n teams from north, n teams from south, => n ties
Draw could go :
N S N S N S S N

Draw 1 Draw 2


S N S N S N N S

Draw 3 Draw 4

P(Draw 1) = P(Draw 2) =P(Draw 3) = P(Draw 4)
= n x n x (n - 1) x (n - 1) x ...
-- ------- -------- --------
2n (2n - 1) (2n - 2) (2n - 3)

= n²!
-----
(2n)!

Let X = n ties of N-S matches. => P(X) = 4(n²!)
-------
(2n!)

If n = 2, P(X) = 4(4) = 16/24 = 2/3
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4!

eg NNSS, NSNS, NSSN => P(X) = 4/6 = 2/3
SSNN, SNSN, SNNS

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JoJo’s snooker balls
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JoJo places various snooker balls in Box 1 & Box 2
(3 of the 8 balls in Box 1 are red ; 4 of the 12 balls in Box 2 are red).
She draws a ball, at random, from both boxes (she calls this a trial), notes the colour and then replaces the balls in their respective boxes. What is the probability that JoJo draws two reds, for the first time only, on trial 3 ?

Solution :
------------
Let n = the number of trials.
Let R(1) = red is drawn in Box 1, R(2) = red is drawn in Box 2.

P[R(1)] = 3/8, P[R(2)] = 4/12 = 1/3

P{R(1) & R(2)}’ = P[R(1) & R(2)’] + P[R(1)’ & R(2)] + P[R(1)’ & R(2)’]
= 1 - P[R(1) & R(2)]


P[R(1) & R(2) in n = 3]
= P{R(1) & R(2)}’ x P{R(1) & R(2)}’ x P[R(1) & R(2)]

= {1 - P[R(1) & R(2)] } x {1 - P[R(1) & R(2)]} x P[R(1) & R(2)]

= {1 - P[R(1) & R(2)]}² x P[R(1) & R(2)]
= {1 - 3/8 x 1/3}² x 3/8 x 1/3
= {1 - 1/8}² x 1/8 = [7/8]² x 1/8 = 49/512 = 0.0957 (4 d.p)

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Lemmy's Aces in a Poker hand
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Calculate the probability of Lemmy getting the Ace of Spades
& the Ace of Clubs in a Poker hand.

Solution :
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Consider the two Aces as a certain entity and we must also discount the other 2 Aces from appearing in the remaining 3 cards.

P(Ace of S & Ace of C) = 1 x 48C3/52C5
= 17,296/2,598,960
= 0.00665 (5 d.p)

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