You wish to fence off a rectangular paddock on one side of a river running through your property in a straight line. No fence is required along the side of the paddock formed by the river. The fence you will use is rolled up in a shed, and you are at the moment not quite sure how long it is. However, you are certain that it is between 3 and 5 km long, and your uncertainty regarding its length can be represented by a parabolic probability density function which tapers off to zero at 3 and 5 kms.
(a) Find and sketch the probability density function of Y, the area of the largest paddock you will be able to fence off.
(b) Find the expected value, mode and median of Y. Then illustrate these three quantities in the figure in (a).
Got an idea:
You require f(3) = f(5) = 0.
Therefore f(y) = a(y - 3)(y - 5).
Integrate f(y) from 3 to 5
Integrate and solve for a.
Once you have the pdf it is simple to calculate E(Y), mode and median.
A rat is released in the space outside a maze consisting of three rooms and six doors, as depicted in the following figure.
Whenever the rat is in a space or room with k doors, it chooses each of these doors to move through next with probability 1/k. We are interested in the movement of the rat from when it first enters the maze until it first leaves.
(a) If the rat enters the maze at Room 1, find the probability that it will leave
from Room 3.
(b) If the rat starts in the space around the maze, find the probability that it will
eventually leave the maze from Room 3.
(c) If the rat leaves the maze from Room 3 find the probability that it entered at
(d) Suppose that the rat is now in the maze and we gain information which
makes us 70% confident that it entered at Room 1 and 20% confident
that it entered at Room 2.
Find the probability that:
(i) the rat will leave from Room 3
(ii) the rat entered at Room 1 if it leaves from Room 3
(iii) the rat entered at Room 1 if it leaves from Room 1.
For (d) it may be assumed that had we known at which room the rat entered the maze,
the said additional information would not alter our beliefs regarding subsequent movements of the rat.
Please help me !!Thank you very much!
You are holding a party at home, and everyone is about to participate in the following game.
Each person will write their name on a card. All the cards will then be collected and randomly redistributed (one per person). If anyone gets back the card with their own name then all the cards will be collected and randomly distributed again, and this process will be repeated until no-one is holding the card with their own name.
When everyone has a card with someone elses name on it, you will call out the name on your card. The called person will then call out the name on their card, and so on, until finally your own name is called out.
If anyones name does not get called out at some stage during this game, they will have to drink a whole 1 litre bottle of vodka by midnight.
(a) Suppose that there are five people at your party (including yourself).
Find the probability that no one will have to drink 1 litre of vodka by midnight.
Then find the expected number of people
who will have to drink 1 litre of vodka by midnight.
(b) Derive general formulas for the probability and expectation in (a), ones which are correct for any number of people attending your party (i.e. 2, 3, 4, etc).
Then apply these two formulas to the cases where there are 2, 3, 4, 5, 10 and 100 people at your party, respectively.
Really need help~