slot machine problem

There is a slot machine consisting of 3 reels with 5 different possible fruits per reel (one of which is a lemon).

a) How many microstates correspond to the macrostate "pictures contain at least one lemon"?

I think it's asking how many different ways lemon can come up out of the 125 possible scenarios.
At first I was thinking 1/5 + 1/5 + 1/5 for .6 of 125 or 75, but I don't think it's right.

b) How many microstates correspond to the macrostate "two pictures are the same"?

First picture 1/5 * 1/5 it will match = 1/25 + 2/5 it will match one of the first two pictures for 44% * 125 = 55

Really not sure how to do this problem and there is nothing in the textbook explaining this, any help is very much appreciated.

Thank you

Hello, dj432!

There is a slot machine consisting of 3 reels with 5 different possible fruits per reel (one of which is a lemon).

a) How many outcomes correspond to "at least one lemon"?

Click to expand...

Each reel can show one of 5 fruits.
There are: .\(\displaystyle 5^3 = 125\) possible outomes.

The opposite of "at least one lemon" is "no lemons".
Each reel must show one of the other 4 fruits.
There are: .\(\displaystyle 4^3 = 64\) outcomes with no lemons.

Therefore, there are: .\(\displaystyle 125 - 64 \:=\:61\) outcmes with at least one lemon.

b) How many outcomes correspond to "two fruit are the same"?

Click to expand...

I will assume this means "exactly two fruit are the same".


We want a matching pair and one other fruit: .\(\displaystyle \{M,M,X\}\)

There are \(\displaystyle 5\) choices for the fruit of the matching pair.

\(\displaystyle \text{The pair can appear in }\,{3\choose2} = 3\text{ ways: }MMX,\:MXM,\:XMM\)

There are 4 choices for the other fruit.


Therefore, there are: .\(\displaystyle 5\cdot3\cdot4 \:=\:60\) outcomes with exactly two matching fruit.

Thank you for the quick response, the way you figured that out makes sense.
Great forum you guys have going here.
Much respect to the time/work you all put in to helping others.