The Leprechaun's Gold

Probability Level 1

A pot has 10 gold coins and 90 iron coins. All the coins are identical to touch.

A lottery with 10 participants is held. Each has a chance to pick just one coin from the bag, without looking. If the coin is gold, the participant will get $10,000. There are no prizes for iron coins.

The 10 participants can mutually agree to pick one of the following strategies to increase the chances of at least one of them getting a prize.

Strategy A : A participant picks up a coin, views it, claims the prize if it is gold and accepts his luck if it is not. He then returns the coin to the bag before thoroughly shuffling it.

Strategy B : Same as Strategy A, however, the picked coin is not returned to the bag.

Which strategy gives a better chance for at least one win?

Both have same chances Strategy B Strategy A

6 solutions

Janardhanan Sivaramakrishnan
Aug 3, 2015

The probability of at least one win is $P(\text{wins} \ge 1) = 1 - P(\text{wins} = 0)$ .

From strategy A , the probability of a drawing an iron coin = $\frac{9}{10}$ .
So, probability of zero wins of 10 participants = $\left(\frac{9}{10}\right)^{10} = 0.3487$ .

So, probability of at least one win through strategy A = $1-0.3487 = 0.6513$ .

For strategy B , the probability of none of the first $n$ participants drawing a gold coin is equals to $\dfrac{90\times89\times\cdots\times(90-n+1)}{100\times99\times\cdots\times(100-n+1)}$

For $n=10$ , this would be $0.3305$ .

Therefore, the probability of at least one win through strategy B = $1-0.3305 = 0.6695$ .

Hence, strategy B is better.

Anibrata Bhattacharya
Aug 3, 2015

Every time someone picks the coin and does not return it, the no of coins in the bag decreases, and even if the coin is not a gold one, the chances of getting a gold coin increases because lesser the no of coins in the bag (including iron and gold ones), the greater are the chances to be able to pick up a gold coin.If an iron coin is picked up, the no of iron coins in the bags decreases as it is removed, thus the odds for getting a gold coin vs an iron coin increases. But, the strategy would be better if a gold coin picked up by chance is returned to the bag.

Anibrata: True enough, but we should also remember that the question asked "Which strategy gives a better chance for at least one win?" (emphasis added). That means we don't need to worry about whether a gold coin is selected at all, only the iron coins, as once a gold coin is selected the conditions of the question are satisfied.

Using logic therefore: whenever an iron coin is selected, the chances of a gold coin being selected increases; whenever a gold coin is selected, satisfies the condition of the question.

Keith Abramowski - 5 years, 10 months ago

Very true. Sorry, didn't bother to read/remember the question properly, after getting the solution correct!

Anibrata Bhattacharya - 5 years, 10 months ago

Andriane Casuga
Aug 10, 2015

Strategy B 10 gold to 90 Iron equals a 1:9 chance. If 1 is removed, it goes to a 1:8.9 chance, if 2 are removed, it goes to a 1:8.8 chance, if 3 are removed, it goes to a 1:8.7 chance... if 9 are remove, the 10th person would have a 1:8.1 chance.

Still not great odds, but better than the original 1:9 chance.

Kyara Riguerra
Nov 17, 2015

They have solutions... I dont have one... I just used logic lol

Hadia Qadir
Aug 12, 2015

Still not great odds, but better than the original 1:9 chance.

Philip Choi
Aug 8, 2015

The possibility of no one getting any prize A: (9^10)/(10^10)~34.9% B: (90/100 89/99 88/98*87/97.....)~ 33.0% Therefore, B is the better option.

The Leprechaun's Gold

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