Date | May 2021 | Marks available | 1 | Reference code | 21M.2.AHL.TZ1.5 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Write down | Question number | 5 | Adapted from | N/A |
Question
Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is 0.8. If it is not sunny, then the probability that it will be sunny the following day is 0.3.
The transition matrix T is used to model this information, where T=(0.8 0.30.2 0.7).
The matrix T can be written as a product of three matrices, PD P-1 , where D is a diagonal matrix.
It is sunny today. Find the probability that it will be sunny in three days’ time.
Find the eigenvalues and eigenvectors of T.
Write down the matrix P.
Write down the matrix D.
Hence find the long-term percentage of sunny days in Vokram.
Markscheme
finding T3 OR use of tree diagram (M1)
T3=(0.65 0.5250.35 0.475)
the probability of sunny in three days’ time is 0.65 A1
[2 marks]
attempt to find eigenvalues (M1)
Note: Any indication that det(T-λI)=0 has been used is sufficient for the (M1).
|0.8-λ 0.30.2 0.7-λ|=(0.8-λ)(0.7-λ)-0.06=0
(λ2-1.5λ+0.5=0)
λ=1, λ=0.5 A1
attempt to find either eigenvector (M1)
0.8x+0.3y=x⇒-0.2x+0.3y=0 so an eigenvector is (32) A1
0.8x+0.3y=0.5x⇒0.3x+0.3y=0 so an eigenvector is (1-1) A1
Note: Accept multiples of the stated eigenvectors.
[5 marks]
P=(3 12 -1) OR P=(1 3-1 2) A1
Note: Examiners should be aware that different, correct, matrices P may be seen.
[1 mark]
D=(1 00 0.5) OR D=(0.5 00 1) A1
Note: P and D must be consistent with each other.
[1 mark]
0.5n→0 (M1)
Dn=(1 00 0) OR Dn=(0 00 1) (A1)
Note: Award A1 only if their Dn corresponds to their P
PDnP-1=(0.6 0.60.4 0.4) (M1)
60% A1
[4 marks]