All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Probability
Mr. Markov plays cards (Posted on 2010-06-26) Difficulty: 3 of 5
Mr. Markov either wins or loses at a card table. His result on a given day is related to his result the day before. If he loses today, there is a 50% chance he would lose tomorrow. If he wins today, there is a 30% chance that he would lose tomorrow. How often would he lose on two consecutive days? Assume an infinite horizon so the first day result is irrelevant. (Note: Losing three days in a row counts as losing on two consecutive days twice)

No Solution Yet Submitted by Andre    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution Comment 1 of 1

First find the probability of a loss; that is, the fraction of the trials (days) that result in a loss:

Let x be that probability (of a loss, not necessarily consecutive).

So, based on the conditional probabilities:

x = .3(1 - x) + .5(x)

Solving this gives x = 3/8.

But each loss is followed by another trial, and in such a trial, 1/2 the time there will be another loss, so this happens 3/16 of the overall trials. (Successes, of course, do not lead to the next day's possibility of being the second day of a two-day loss pair.) So overall, 3/8 of the time there is a loss, and in 1/2 of those times, another loss follows--3/16 of the days are loss days that follow after at least one other loss day.

Therefore 3/16 is the answer.

Simulation verification:

 trials = trials + 1
 r = RND(1)
 IF prevLoss THEN
    IF r < .5 THEN loss = 1:  ELSE loss = 0
    IF loss THEN dblLoss = dblLoss + 1
    IF r < .3 THEN loss = 1:  ELSE loss = 0
 prevLoss = loss
 IF loss THEN losses = losses + 1
 PRINT losses, dblLoss, trials, dblLoss / trials

results, as the last line, when stopped, in:

 losses      cons losses     trials             prob
 161143        80562         428453        .1880299589453219

  Posted by Charlie on 2010-06-26 14:58:43
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information