Question

Gertrude will open a movie theatre in the town of Northern Millstone on July 20th, 2020. Assume the following to be true.

If a resident of Northern Millstone patronizes Gertrude’s movie theatre on any given day, the probability that this resident will patronize Gertrude’s movie theatre the next day is .05.

If a resident of Northern Millstone does not patronizeGertrude’s movie theatre on any given day, the probability that this resident will not patronize Gertrude’s movie theatre the next day is .9.

If 10% of the residents of Northern Millstone patronizeGertrude’s movie theatre on July 20th, 2020, what percent of the residents of Northern Millstone will patronize Gertrude’smovie theatre on July 22nd, 2020?

You can use a computer or a calculator for the matrix multiplication. I think the answer already existing here is wrong.

Answer 1

The initial probability distribution is given by v.

The transition matrix P is given by

Now we will write the initial probability of distribution.

Probability distribution on 20th July 2020 is

Probability distribution on 21st July 2020 is

Probability distribution on 22nd July 2020 will be

Hence, the percentage of residents patronizing the movie theatre on 22nd July.

Hence, 9.525% of the residents will patronize the movie theatre on 22nd July 2020.

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