In: Statistics and Probability
Commercial fishermen in Alaska go into the Bering Sea
to catch all they can of a
particular species (salmon, herring, etc.) during a restricted
season of a few weeks.
The schools of fish move about in a way that is very difficult to
predict, so the fishing in a
particular spot might be excellent one day and terrible the
next.
The day-to-day records of catch size were used to discover that the
probability of a good
fishing day being followed by another good day is 0.5, by a medium
day 0.3, and by a poor
day 0.2.
A medium day is most likely to be followed by another medium day,
with a probability of 0.4,
and equally likely to be followed by a good or bad day.
A bad day has a 0.1 probability of being followed by a good day,
0.4 of being followed by a
medium day, and 0.5 probability of being followed by another bad
day.
a) (10 points) If the fishing day is bad on Monday, what is the
probability that it will be
medium on Thursday?
b) (10 points) Suppose the fishing day will be good w.p. 0.25,
medium w.p. 0.30 and bad
with 0.45 on the current day, which is a Tuesday. How do you think
fishermen came up
with these probabilities for the current day? Argue.
c) (10 points) Given the probabilities in part b, calculate the
probability of having a bad
fishing day after three days.
d) (10 points) What is the probability of four consecutive good fishing
days until it gets
worse.
e) (10 points) If the fishing day is medium initially, for how many
days on the average will it
remain medium? What is the distribution of this number of days?
Answer:
Given that,
Commercial fishermen in Alaska go into the Bering Sea to catch all they can of a particular species (salmon, herring, etc.) during a restricted season of a few weeks.
The schools of fish move about in a way that is very difficult to predict, so the fishing in a particular spot might be excellent one day and terrible the next.
The day-to-day records of catch size were used to discover that the probability of a good fishing day is followed by another good day is 0.5, by a medium day 0.3, and by a poor day 0.2.
A medium day is most likely to be followed by another medium day, with a probability of 0.4, and equally likely to be followed by a good or bad day.
A bad day has a 0.1 probability of being followed by a good day, 0.4 of being followed by a medium day, and 0.5 probability of being followed by another bad day.
Let us list all the probabilities.
P(G|G) = probability of a good day is followed by another good day = 0.5
P(M|G) = probability of a good day is followed by a medium day = 0.3
P(P|G) = probability of a good day is followed by a poor day = 0.2
P(G|M) = probability of a medium day being followed by a good day = 0.1
P(M|M) = probability of a medium day being followed by another medium day = 0.4
P(P|M) = probability of a medium day being followed by a poor day = 0.1
P(G|P) = probability of a poor day being followed by a good day = 0.1
P(M|P) = probability of a poor day being followed by a medium day = 0.4
P(P|P) = probability of a poor day being followed by another poor day = 0.5
(a).
If the fishing
day is bad on Monday, what is the probability that it will be
medium on Thursday:
The transition probabilities can be expressed as a matrix,
P =
G | M | P | |
G | 0.5 | 0.3 | 0.2 |
M | 0.1 | 0.4 | 0.1 |
P | 0.1 | 0.4 | 0.5 |
The transition probabilities for Monday, given that today is Thursday is
=P PP
=P^2P
P^2=
G | M | P | |
G | 0.3 | 0.35 | 0.23 |
M | 0.1 | 0.23 | 0.11 |
P | 0.14 | 0.39 | 0.31 |
And then,
P^2P=
G | M | P | |
G | 0.208 | 0.322 | 0.21 |
M | 0.084 | 0.166 | 0.098 |
P | 0.14 | 0.322 | 0.222 |
Doing matrix multiplication, we obtain the transition probabilities for Thursday as:
G | M | P | |
G | 0.208 | 0.322 | 0.21 |
M | 0.084 | 0.166 | 0.098 |
P | 0.14 | 0.322 | 0.222 |
The probability that Thursday is a medium day, given that today is a poor day
= P(M|P) in the above matrix
= 0.322
(b).
Suppose the fishing day will be good w.p. 0.25, medium w.p. 0.30 and bad with 0.45 on the current day, which is a Tuesday. How do you think fishermen came up with these probabilities for the current day? Argue:
The fishermen would have observed data over a long period of time to come up with these probabilities.
(c).
Given the
probabilities in part b, calculate the probability of having a
bad
fishing day after three days.
The transition matrix for 3 day-period is given as P PP =
G | M | P | |
G | 0.208 | 0.322 | 0.21 |
M | 0.084 | 0.166 | 0.098 |
P | 0.14 | 0.322 | 0.222 |
P(P)= P(G) P(P|G)+P(M) P(P|M)+P(P) (P|P)
=0.25 0.21+0.30 0.098+0.45 0.222
=0.0525+0.0294+0.0999
=0.1818
(d).
What is the
probability of four consecutive good fishing days until it
gets
worse:
Probability of 4 consecutive good days followed by a medium or poor day
= 2[P(G|G) [P(P|G) + P(M|G)]]
=2[0.5 [0.2+0.3] ]
=2[ 0.25]
=0.5
(e).
If the fishing day is medium initially, for how many days on the average will it remain medium? What is the distribution of this number of days?
If a particular day is medium, probability that the next day is better = 0.1+0.5 = 0.6 and probability next day is medium= 0.4
This is a binomial distribution with probability of success (Medium day) = 0.4 and probability of failur = 0.6
Expected value of n, where n is the number of days it will be bad consecutively, given that first day is bad
= 2 x 0.4 x 0.6 + 3 x 0.41 x 0.6 + 4 x 0.42 x 0.6 + 5 x 0.43x 0.6 + ...
=0.6 x [ 2 x 0.4 + 3 x 0.41 + 4 x 0.42+ 5 x 0.43 + ...]
We know the series 1 + 2x + 3x2 + 4x3 + ... =
Our required expected number of days is,
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