Question

A medical researcher is studying the spread of a virus in a population of 1000 laboratory mice. During any week, there is a 70% probability that an infected mouse will overcome the virus, and during the same week there is a 40% probability that a noninfected mouse will become infected. Four hundred mice are currently infected with the virus. How many will be infected next week and in 3 weeks?

(a) next week mice rilex
(b) in 3 weeks mice

Answer 1

a) 360 mice

b) 364 mice

Explanation:

Assume :

A = number of infected mice , B = number of non-infected mice

P( infected mice overcoming infection ) = 70% = 0.7

P ( Infected mice not overcoming infection ) = 1 - 0.7 = 0.3

P( mice becoming infected ) = 40% = 0.4

P ( mice not becoming infected ) = 1 - 0.4 = 0.6

Number of infected mice = 400

Number of non-infected mice = 1000 - 400 = 600

step 1 ; express the probabilities in matrix form

`P = \left[\begin{array}{ccc}0.3&0.4&\\0.7&0.6&\\\end{array}\right]`

`X = \left[\begin{array}{ccc}400\\600\\\end{array}\right]`

step 2 : multiply the matrix above to determine the number of mice that will be infected

a) For next week

PX =
`\left[\begin{array}{ccc}360\\640\\\end{array}\right]`

i.e. 360 mice will get infected next week

next 2 week = P ( PX )

=
`\left[\begin{array}{ccc}0.3&0.4&\\0.7&0.6&\\\end{array}\right]` *
`\left[\begin{array}{ccc}360\\640\\\end{array}\right]` =
`\left[\begin{array}{ccc}364\\636\\\end{array}\right]`

b) In 3 weeks time

P ( P(PX) =
`\left[\begin{array}{ccc}0.3&0.4&\\0.7&0.6&\\\end{array}\right]` *
`\left[\begin{array}{ccc}364\\636\\\end{array}\right]`

=
`\left[\begin{array}{ccc}363.6\\636.4\\\end{array}\right]`

i.e. 364 mice will get infected in 3 weeks time