Math Is Fun Forum

TomBrady12

Member

Registered: 2008-02-06

Posts: 1

Probability problem

Could use some help with this:

Karen and Diane are having an argument.  They both know that a gumball machine is loaded with gumballs of 6 different colors. Karen feels lucky.  She thinks she can get each color by using only 6 pennies.  Diane feels it will take at least 15.  Let's assume that the machine is loaded with an equal number of each color.  How many pennies do you think are needed to get one of each color?  Do you agree with either Karen or Diane?

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Probability problem

Diane is completely wrong.  Karen is kind of right; she can get all 6 colors with just 6 pennies, it's just not likely.  In order to guarantee that you'd get one of every color you would need (5/6)*n + 1 pennies, where n is the total number of gumballs in the machine.

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Wrap it in bacon

TomBrady12

Member

Registered: 2008-02-06

Posts: 1

Re: Probability problem

.........................................what?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Probability problem

I get the impresion the question is more like this way: What is the minimum number of pennies you have to spend so that the probability of getting one of each colour is at least half?

Obviously you’ll need to spend at least 6 pennies. Suppose you spend k pennies. Now let’s work out the probability of not getting at least one of each colour. Label the colours A–F.

Probability of not getting colour A =

Same probability of not getting colours B–F.

Probability of not getting colour A or B =

Same probability of not getting any two specified colours.

Probability of not getting colour A or B or C =

Same probability of not getting any three specified colours.

And so on, till we get probability of not getting any five specified colours =

Now we use this formula:

So the probability of not getting at least one of each colour is

So the probability of getting at least one of each colour is

.

Using Excel I find the minimum k for which

to be 13. So you need to spend at least 13 pennies for it to be at least 50% likely to get what you want.

Last edited by JaneFairfax (2008-02-07 21:32:43)