Question

Louie walks to and from work every day. In his city, there is a 50 percent chance of rain each morning and an independent 40 percent chance each evening. His habit is to bring (and use) an umbrella if it’s raining when he leaves the house or office, but to leave them all behind if not. Louie owns three umbrellas.

On Sunday night, two are with him at home and one is at his office. Assuming it never starts raining during his walk to his home or office, what is the probability that he makes it through the work week without getting wet?"

Answer 1

This problem can be solved by forming a Markov chain. Specifically, it has nine states, defined as follows:

% 1: at home, with 0 umbrellas

% 2: at work, with 3 umbrellas

% 3: at home, with 1 umbrellas

% 4: at work, with 2 umbrellas

% 5: at home, with 2 umbrellas

% 6: at work, with 1 umbrellas

% 7: at home, with 3 umbrellas

% 8: at work, with 0 umbrellas

% 9: gets wet

The transition matrix P has element P(i,j) = probability of transitioning from state i to state j. It is as follows:

Another way to see this is that the rules for forming the matrix is

% if at home, move from

%    n to n+1, if it does not rain (prob. 0.5)

P(1,2) = P(3,4) = P(5,6) = P(7,8) = 0.5

%    n to n-1, if n> 1 and to 9 if n==1, and it does rain (prob. 0.5)

P(1,9) = P(3,2) = P(5,4) = P(7,6) = 0.5

% if at work, move from

%   n to n-1, if it does not rain (0.6)

P(2,1) = P(4,3) = P(6,5) = P(8,7) = 0.6

%   n to n+1, if it does rain (0.4)

P(2,3) = P(4,5) = P(6,7) = P(8,9) = 0.4

% once you get wet, you’re wet

P(9,9) = 1

The starting point is state 5 (at home with two umbrellas), represented by the following row vector:

pstart=[0,0,0,0,1,0,0,0,0].

The probability vector after one trip is

pstart⋅P=[0,0,0,0.5,0,0.5,0,0,0]

So, we are at the office with 2 umbrellas if it rained in the morning (with probability 0.5), and at the office with 1 umbrella if it did not (also with probability 0.5).

The probability vector after the five-day work week (ten trips) is computed by

pend=pstart⋅P10=[0.1120,0,0.2291,0,0.2255,0,0.1261,0,0.3073]

So, pwet=0.3073.

and pdry=1−pwet=0.6927