Oxford University MAT 2021 Solved Papers

Question : 5 of 10

Marks: +1, -0

Six vectors

v_1, v_2, v_3, v_4, v_5, v_6

are each chosen to be either

(\table 1 ; 1)

(\table 3 ; 2)

with equal probability, with each choice made independently. The probability that the sum

v_1+v_2+v_3+v_4+v_5+v_6

is equal to the vector

(\table 10 ; 8)

Solution:

In order to make the vector

(\table 10 ; 8)

we would need

a(\table 1 ; 1)+b(\table 3 ; 2)=(\table 10 ; 8)

where

a

is the number of times we pick

(\table 1 ; 1)

and

b

is the number of times we pick

(\table 3 ; 2)

.
Since we have six vectors,

a+b=6

. Solving the simultaneous equations

a+3 b=10

and

a+2 b=8

, we get

a=4

and

b=2

, and we can check that

a+b=6

for this solution! So we want exactly two of the six vectors to be

(\table 3 ; 2)

. There are

{ }^6 C_2=15

ways that this could happen, each with probability

\;{1}/{64}

, so the answer is

\;{15}/{64}

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