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Okay so I am designing a probability game involving three spinners called "WordWheel."

There are a few simple rules for the game:

  1. There will be 3 spinners that will each have different letters on them. The first spinner will have B,T,A,E,S. The middle spinner will have A,E,I,O,U.The last spinner will have M,R,Y,D,J.
  2. The whole point in this game is to get an actual three letter word.
  3. Before you spin, you have to bet a value of tokens (you start off with 5) on getting a word with a specific letter in it.
  4. Once all three have been spun, if you don’t get a word with that letter, you loose the bet.

Now, I already know that there is a total of 125 different outcomes using a tree diagram. I have also looked into which of those are ACTUAL words, which ends up in 20. The only ones I actually classified as real words is: Bay, Bad, Boy, Bud, Bar, Bam, Bed, Bid, Bum, Toy, Tad, Tar, Air, Aim, Aid, Ear, Sod, Sum, Sad, Say. (and I would like to keep it this way, makes it easier for me!)

So I figured it would be P(win)= 20/125

But I also have to incorporate the bet with a specific letter in it. So I thought I should find how many times each letter occurs out of the 20 words, shown below.

P(B)= 9/20 , P(T)= 3/2 , P(A)= 12/20 , P(E= 2/20 , P(S)= 4/20 , P(I)= 4/20 P(O)= 3/20 , P(U= 3/20 , P(J)= 0/20 , P(D)= 8/20 , P(Y)= 3/20 , P(R)= 4/20 P(M)= 4/20

I just don't know what to do with these to find out the final probability of winning.

Those are just my ideas, so I do realize I may not be on the right track but if I am wrong please correct me and help me to get in the right direction. I'm really hoping this game works.

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From your description, I understand that to win you have to satisfy two conditions simultaneously, i.e., a) you get "any" actual word out of those possible 20 words you mentioned and b) you get that specific letter in that word on which you placed your bet.

If my understanding of your game is correct then you need to identify two separate events. Let $E_1$ is the event that an actual word is found, let $E_2$ be the event that the letter you placed your bet on is found when you spin the wheels.

Now you need the probability $P(E_1 \cap E_2)$. From the definition of conditional probability, we know that $P(E_1 \cap E_2) = P(E_2|E_1)P(E_1)$.

So lets assume I place my bet on letter A. Then if it is given that $E_1$ has occurred, i.e., if we know that a proper word has been found then $P(E_2=A|E1) = 12/20$, as you calculated above.

Also you have calculated that $P(E_1)=20/125$, so the probability of winning if I place my bet on A is $12/20 \times 20/125 = 12/125$, which is quite apparent actually.

Also since A occurs the most in your actual words so everyone should place their bets on A to maximize their chances of winning.

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