Leaving Certificate Mathematics – Higher Level
Question type
Level
Section B
LCHL
Strand 2
A ball is dropped onto peg A of a cascade toy, illustrated below. At each peg, the ball can
fall either to the left or the right on its way down the cascade, until it falls into one of the
numbered collection bins at the bottom. Possible paths to point G at level 3 are shown.
a) To hit peg G (at level 3) there are three different paths that the ball can take on the
way down from A: BDG, BEG, or CEG.
How many different paths can the ball follow so that it will land on
(i)
peg F?
(ii)
level 3?
b) What is the probability that a ball, dropped on peg A, will land on peg H?
Leaving Certificate Mathematics – Higher Level
c) To get to peg J from A, the ball must hit four pegs, falling to the left three times
and to the right once. One sequence could be Left, Left, Left, Right.
(i)
Verify that the number of different paths to reach peg J is given by 4 C1
How many different paths are there for the ball to get to
(ii)
peg K?
(iii)
collection bin 3?
d) The probability of falling to the left (or right) is 0.5 for each peg. What is the
probability that a ball, dropped on peg A, will fall into
(i)
collection bin 1?
(ii)
collection bin 3?
Leaving Certificate Mathematics – Higher Level
e) The cascade toy is used at a stall in a charity fund-raising campaign. A player
pays 1€ for a ball to be dropped on peg A and is paid the amount listed below for
the bin where it finishes.
Prize
Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
Bin 6
€2
€1
50c
50c
€1
€2
Probability
Expected
value
(i)
Complete the table to show the probabilities and expected payout values
for the six collection bins.
(ii)
If the game is played 1000 times, what profit can the stall operator expect
to make?
Leaving Certificate Mathematics – Higher Level
Leaving Certificate Mathematics – Higher Level
Solution
(a)
(i) 1 (path BDF)
No partial credit.
(ii) 8 (1 + 3 + 3 + 1)
Partial credit for 1 + 3 + 3 = 7.
(b)
3
or 0.375
8
(c)
(i) To get to K one R is needed from these 4 pegs. Hence 4 C1 .
Full marks for result consistent with answers at (a) above.
Accept listing of the 4 possible paths to K and stating that 4 C1 = 4.
Partial credit for stating 4 C1 = 4 or listing the 4 paths, but no explanation.
(ii) 6 or 4 C 2 .
Partial credit for at least 4 correct paths listed.
(iii) 10 or 5 C 2
Partial credit for stating 5 steps (or pegs) with 2 R needed
but no calculation, or correct listing of at least 7 paths.
(d)
(i)
1
or 0.03125
32
High partial credit for correct expression not worked out,
or for correct total of 32 paths (1, 5, 10, 10, 5, 1) but prob.
not completed.
Low partial credit for a suitable probability value derived
from work shown.
(ii)
10
or 0.3125
32
High partial credit for all work shown, but one error.
Mid partial credit for correct total number of paths [32]
but probability not completed.
Low partial credit for at least two correct calculation steps
leading to a probability calculation.
(e)
(i)
Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
Bin 6
Prize
€2
€1
50c
50c
€1
€2
Probability
0.03125
0.15625
0.3125
0.3125
0.15625
0.03125
Expected
value
6.25c
15.625c
15.625c
15.625c
15.625c
6.25c
Leaving Certificate Mathematics – Higher Level
Full credit for equivalent fraction values.
High partial credit for substantially correct table values.
Mid partial credit for correct probability values, or
expected values calculated appropriately from incorrect
probability values.
Low partial credit for at least three correct probability
values and some attempt to calculate expected values.
(ii)
€250 [75c  1000 = €750 payout; income = €1000 €250 profit]
Full credit for correct working of values supplied in
candidate’s answer to part (i)
High partial credit for correct work but with an error, or
stops at €750 payout.
Low partial credit for correct total of expected values in
table, but fails to finish.