250 adults are practicing every day before taking part in a high-stakes children's game. The game requires them carefully to remove a shape- either a star or an umbrella-from a sugar wafer. 30% of those who choose stars on a given day will switch to umbrellas the next day, and 20% of those who choose umbrellas on a given day will switch to stars the next day. (a) If 150 contestants practice with stars today, how many will practice with umbrellas tomorrow? (b) If the actual game is held a long time from now, what proportion of contestants do we expect will be practicing with stars each day close to then?
250 adults are practicing every day before taking part in a high-stakes children's game. The game requires them carefully to remove a shape- either a star or an umbrella-from a sugar wafer. 30% of those who choose stars on a given day will switch to umbrellas the next day, and 20% of those who choose umbrellas on a given day will switch to stars the next day. (a) If 150 contestants practice with stars today, how many will practice with umbrellas tomorrow? (b) If the actual game is held a long time from now, what proportion of contestants do we expect will be practicing with stars each day close to then?
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.6: Summarizing Categorical Data
Problem 13CYU
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Transcribed Image Text:250 adults are practicing every day before taking part in a high-stakes children's game. The game requires
them carefully to remove a shape- either a star or an umbrella-from a sugar wafer. 30% of those who choose
stars on a given day will switch to umbrellas the next day, and 20% of those who choose umbrellas on a
given day will switch to stars the next day.
(a) If 150 contestants practice with stars today, how many will practice with umbrellas tomorrow?
(b) If the actual game is held a long time from now, what proportion of contestants do we expect will be
practicing with stars each day close to then?
![Answer
The corresponding Markov matrix is P
=
[0.7 0.2]
0.8
(If you choose to list umbrellas first and stars
0.3
[0.8 0.3]
second, then there is one other correct matrix: P =
No other matrix is correct.) Using this
0.2 0.7
matrix, we find for part (a) that 125 are practicing with umbrellas tomorrow. For part (b), the unique steady-
state vector (eigenvector of P with eigenvalue λ = 1 whose entries are positive and sum to 1) is
in the long run, 2/5 (or 40%) will be practicing with stars.
[2/5]
SO
3/5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Feaf5f93f-982f-4ff1-90c4-a262552203d2%2Fmdh1pcq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Answer
The corresponding Markov matrix is P
=
[0.7 0.2]
0.8
(If you choose to list umbrellas first and stars
0.3
[0.8 0.3]
second, then there is one other correct matrix: P =
No other matrix is correct.) Using this
0.2 0.7
matrix, we find for part (a) that 125 are practicing with umbrellas tomorrow. For part (b), the unique steady-
state vector (eigenvector of P with eigenvalue λ = 1 whose entries are positive and sum to 1) is
in the long run, 2/5 (or 40%) will be practicing with stars.
[2/5]
SO
3/5
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