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Yesterday you had 50 coins totaling exactly one US dollar in value. Somehow you lost one of those fifty coins on the way to the vending machine. Assume the likelihood of losing each coin (regardless of its physical size) is the same.

What is the likelihood that the coin you lost is a nickel?

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  • $\begingroup$ Is the half-dollar (50 cents) a coin that can be one of the 50? $\endgroup$
    – mbjb
    Commented Jun 10, 2016 at 3:19
  • $\begingroup$ @mestackoverflow It can be if you can find a valid combination of 50 coins than includes it and totals $1 USD. Good luck with that:) $\endgroup$
    – K Chan
    Commented Jun 10, 2016 at 3:21

2 Answers 2

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There are $2$ ways to make a dollar with exactly $50$ coins.
- $45$ pennies, $1$ quarter, $2$ dimes, and $2$ nickels
- $40$ pennies, $8$ nickels, and $2$ dimes

In the first combination, there is a $4\%$ chance that the lost coin was a nickel $(2/50 = 0.04 = 4\%);$ in the second, there is a $16\%$ chance of the lost coin being a nickel $(8/50 = 0.16 = 16\%)$. Combined, the percentages equal $20\%$ out of $200\%$; therefore, the final probability is $10\%$.

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    $\begingroup$ But what if I'm a coin collector and I was carrying fifty two-cent pieces? $\endgroup$
    – f''
    Commented Jun 10, 2016 at 3:47
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    $\begingroup$ Well, you wouldn't want to use them in the vending machine, would you?... $\endgroup$
    – mbjb
    Commented Jun 10, 2016 at 3:58
  • $\begingroup$ And the machine probably wouldn't accept anyone 2-cent coins anyway. Good idea, though. $\endgroup$
    – mbjb
    Commented Jun 10, 2016 at 3:59
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    $\begingroup$ Of course the machine wouldn't take the two-cent pieces, that's why I was also carrying some $1 bills. :P $\endgroup$
    – f''
    Commented Jun 10, 2016 at 4:06
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    $\begingroup$ The last step in this answer just assumes that both configurations are equally likely which is not necessarily true. There doesn't seem to be enough information in the question to meaningfully combine the results for each configuration. $\endgroup$
    – theosza
    Commented Jun 10, 2016 at 9:21
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Answer:

Probability = 0.16 = 8/50

Coins originally in pocket:

2 dimes, 8 nickels, 40 pennies = 20c + 40c + 40c = $1

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    $\begingroup$ That would be the correct answer if that was the only combination of 50 coins that added up to $1USD $\endgroup$
    – K Chan
    Commented Jun 10, 2016 at 3:39
  • $\begingroup$ Ah, yes. I couldn't find that one. $\endgroup$
    – Paul Beckingham
    Commented Jun 10, 2016 at 3:49

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