slot machine problem

[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 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

 

[soroban]

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"?

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"?

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.

 

[dj432]

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.