please help: A math-olympics is being held this weekend among SPC Math professors....

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steenson

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A math-olympics is being held this weekend among SPC Math professors. There are 30 professors signed up, and the first 3 finishers will receive $100, $50, and $20, respectively. If we assume that each professor has an equal chance of winning, what is the probability that Prof. Weideman will finish second?

Thank you in advance
with steps to help me better understand
​
 
steenson said:
A math-olympics is being held this weekend among SPC Math professors. There are 30 professors signed up, and the first 3 finishers will receive $100, $50, and $20, respectively. If we assume that each professor has an equal chance of winning, what is the probability that Prof. Weideman will finish second?

Thank you in advance
with steps to help me better understand
​
Click to expand...

How many ways are there to choose 3 people to get the prizes?

In how many of those is Prof. Weideman in second place?

Or, to make it even easier:

How many ways are there to choose the second place winner?

How many of those are Prof. Weideman?

Try both ways, and see that they agree, which surprises some people.
 
steenson said:
A math-olympics is being held this weekend among SPC Math professors. There are 30 professors signed up, and the first 3 finishers will receive $100, $50, and $20, respectively. If we assume that each professor has an equal chance of winning, what is the probability that Prof. Weideman will finish second?​
Click to expand...
There are \(\displaystyle _{30}\mathscr{P}_3=24360 \) ways to permute three of thirty(SEE HERE)
Now there are \(\displaystyle 29\cdot 28 \) ways to have \(\displaystyle F{\bf W}T \) i.e. Prof W as second place.
Can you now answer?
 
Last edited:
pka said:
There are \(\displaystyle _{30}\mathscr{P}_3=24360 \) ways to permute three of thirty(SEE HERE)
Now there are \(\displaystyle 30\cdot 28 \) ways to have \(\displaystyle F{\bf W}T \) i.e. Prof W as second place.
Can you now answer?
Click to expand...

Are you sure? Why 30*28?
 
steenson said:
A math-olympics is being held this weekend among SPC Math professors. There are 30 professors signed up, and the first 3 finishers will receive $100, $50, and $20, respectively. If we assume that each professor has an equal chance of winning, what is the probability that Prof. Weideman will finish second?
Click to expand...

Dunno...but something doesn't feel right with that problem...
why mention the prize money?
Isn't it the same as 30 people picking at random from a box
containing 30 balls numbered 1 to 30?
 
Denis said:
Dunno...but something doesn't feel right with that problem...
why mention the prize money?
Isn't it the same as 30 people picking at random from a box
containing 30 balls numbered 1 to 30?
Click to expand...
They just formally numbered the first three but not the last 27. Does it really matter that they did not number the last 27?--after all they are losers
 
Denis said:
Dunno...but something doesn't feel right with that problem...
why mention the prize money?
Isn't it the same as 30 people picking at random from a box
containing 30 balls numbered 1 to 30?
Click to expand...

There's nothing wrong with a problem giving information that is not needed; isn't that what real world problems are all about? The point is to choose what matters. And here, I think that's the main point of the problem.

And you are exactly right: if we focus on only what really matters, the problem is very simple. And reframing a probability problem in terms of picking balls or cards or rolling dice can be very helpful in removing the extraneous material. All that matters is whether the given person gets the ball numbered 2; what others get, or who gets the other balls, is irrelevant.
 
Dr.Peterson said:
There's nothing wrong with a problem giving information that is not needed; isn't that what real world problems are all about? The point is to choose what matters. And here, I think that's the main point of the problem.

And you are exactly right: if we focus on only what really matters, the problem is very simple. And reframing a probability problem in terms of picking balls or cards or rolling dice can be very helpful in removing the extraneous material. All that matters is whether the given person gets the ball numbered 2; what others get, or who gets the other balls, is irrelevant.
Click to expand...
In the real world, what usually happens is that you are initially given a whole lot of information, much of it irrelevant and some of it wrong, but not given all the information required to answer the question posed, which is not the question intended.

Fortunately, few text books try to mimic the real world.
 
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