One Billion Pound Raffle Part 1
There's going to be a raffle for one billion pounds (UK currency for those who don't know ^^) in a giant room.
There's a bag filled with 50 marbles, one for each person in the room, 49 black, and only one red.
As you might have already guessed, the person that picks the red marble out of the bag will get one billion pounds.
Now, you are among the 50 people in the room.
There will be a line of people drawing a marble out of the bag (no replacements).
Which place in line (like 1st, 2nd, 3rd, etc., to draw a marble) is most likely to win the prize of one billion pounds?
***Remember your answers for Part 2***
There's a bag filled with 50 marbles, one for each person in the room, 49 black, and only one red.
As you might have already guessed, the person that picks the red marble out of the bag will get one billion pounds.
Now, you are among the 50 people in the room.
There will be a line of people drawing a marble out of the bag (no replacements).
Which place in line (like 1st, 2nd, 3rd, etc., to draw a marble) is most likely to win the prize of one billion pounds?
***Remember your answers for Part 2***
posted by James Yu at
8:08 PM
18 Comments:
It makes no difference where you stand in line.
The first person has a 1/50 chance.
The second person has a 49/50 chance of missing out, times a 1/49 chance of winning = 1/50.
Likewise the third person has a 48/50 x 1/48 = 1/50 chance.
And so on down the line. The last person has a 49/50 chance of missing out = a 1/50 chance of winning.
Everyone has a 1/50 chance.
If each marble is revealed as it drawn, it may seem different than
if every marble is hidden till the end, it really does not change the odds.
One billion pounds? I would guess there would be a good chance of a hold up before the last marble was drawn. Ha!
All places in line have a 1/50 chance.
(However - just to play Maxwell's Demon here - the 50th place has the "most likelihood at his time of pulling" if the raffle is ended immediately when the red marble is pulled. In that scenario, if the 50th person actually makes it to his chance of pulling from the bag; he has a 100% chance of winning.)
The 50th person has the best chance. He has a 100% chance. Remember, we are talking about chances here; without replacing the ball. So, the first person has 1/50 chance, 2nd person has 1/49 chance and so on. But, the last person has a 1/1 chance.
Yes the last person has a 1/1 chance of picking the right marble given that no one else picked it already, therefore i don't think its the last one, i'm thinking its some one around 25 that will have the best odds
#42!
first person to draw a gun and kill veryone else
Odds are it won't be drawn first, and as the total gets lower, the odds increase.
Odds are it will be drawn before the 50th person as well.
So I would assume a bell curve would apply. I'd like to be in #25 or #26 position for that lineup.
I agree with BowlingTracker. So who is right?
The first one is most probably going to take the red ball, unless he is really stupid.
That is so because although everyone assumed it's a blind draw, it was never *stated*.
Everyone has an equal chance of drawing it - 1/50.
The 1st person has a 1/50 chance of picking the ball. He is also knows he will get to draw, he has a 50/50 chance of drawing. Therefore 1/50 x 1/1 = 1/50
The 50th person has a 1/1 chance of picking it if he gets to draw, but there is a 1/50 chance that he will even get to draw, (49 people in front of him already attempted to draw). Thus a 1/1 x 1/50 = 1/50 chance.
The 25th person has a 1/26 chance of picking it if he gets to draw. But he only has a 26/50 chance of drawing (there are 24 people in front of him, thus 50 balls minus 24 chances of the ball being drawn equals 26). This means 1/26 x 26/50 = 1/50 chance. You can do this with all spots on the list.
So statistically that proves everyone has an equal chance. Now whether someone feels lucky at the front, middle, or end is up to them.
1st
it never says that you cant look at the marble of your choice. The first person can see the red marble and win.
I intuitively believed that it didn't matter what spot in line you took, but I wrote a sample program simulation and indeed every spot in line has an equal chance of winning
I would rephrase this problem into the following questions: "If an 'investor' could purchase each person's ticket (i.e. place in line), one at a time before each person draws, what is the fair value of each ticket? Which ticket has the highest fair value?" (Recall that "fair value" does not allow the investor to make a riskless profit.)
If the probabilities are equal, the investor would approach the first person in line and offer 40 million (= 2 billion prize * 1/50 probability) to take his place, draw a losing black ball, then offer 40 million to the second person, and so on. If the red ball is drawn last, he would have spent 2 billion and collected the 2 billion prize and break even. However, if the red ball is drawn any time before that, he would have spent less than 2 billion and made a riskless profit. This suggests that the "fair value" of the earlier tickets is higher, while the value of the later tickets drops to zero after the red ball is drawn.
However, the ticket-holder will demand a higher price for his ticket toward the end. The last person will demand 2 billion if the red ball is the only one remaining. The 49th person has a 50-50 chance, so he might demand 1 billion, and so on, although there is no guarantee that everyone will get a chance to draw.
Therefore, this problem is a paradox because even though the probabilities are equal, each ticket has different values from the perspective of the investor and that of the ticket-holder.
Oops, I thought it was a 2 billion raffle for some reason. For this 1 billion raffle, please divide all my numbers by 2 in my previous comment.
its easy they did not say u can not see the balls so first just pick the red 1 ....us geordies r awesome no flys on us eh!!
the one who kills everyone then runs away
Ray, the problem isn't a paradox because you comparing two different things. You agreed that everyone has the same odds, and then you are trying to give each ticket a value right before they draw. As you go down the line, the odds for the person to get the right ball go up right before the draw (if it hasn't been drawn). If the investor had to buy a ticket before everyone started, they would all be worth the same price, thus they all have the same odds.
No paradox here.
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