Draw me until you match me

Main post link -> https://brilliant.org/mathematics-problem/draw-me-until-you-match-me/?group=NKp1aUV3OfuM

I want nobody to tell the correct answer here.I already bought the discussion and the answer seems wrong to me. I just wanted to know what is wrong with this approach:

First, let us calculate total number of such possible ways ,

Case-1 : We get the same numbers in first and second turns,

For the first turn, we have 4 options and for the second turn , we have only one option, hence total number of ways = 44.

Case-2 : First 2 are different and the third is same as either first or second.

Hence, first turn has 4 options, second turn has 3 options and third turn has 2 options(either same as first turn or same as second turn). Hence no. of ways = 2424.

Case-3 : First three numbers are different and fourth one is same as either of them.

First turn has 4, second turn has 3, third turn has 2 and fourth turn has 3 options, hence 7272 ways.

Case-4 : First four turns are different and the fifth turn is either of them.(Note that fifth turn has 4 options).

Number of ways = 4!×44! \times 4 = 9696

Hence total number of ways = 196196

Now, let us find no. of favourable ways. Note that now, in every case last turn will have only 11 option, as it is same as the first one. Hence everytime i will be multiplying by 1 showing last turn has 1 option only.

Case - 1: First 2 turns are same. 44 ways.(every possible way will satisfy)

Case-2 : First 2 turns are different and third turn is same as first.

4×3×1=124 \times 3 \times 1 = 12 ways

Case - 3: First 3 turns are different and fourth one is same as the first one :

4×3×2×1=244 \times 3 \times 2 \times 1 = 24 ways.

Case-4 : First four turns are different and fifth turn is same as first turn :

4×3×2×1×1=244 \times 3 \times 2 \times 1 \times 1 = 24 ways.

Hence total number of favorable ways = 6464.

Hence probability = 64196=1649.\frac{64}{196} = \boxed{\frac{16}{49}} .

Note by Jatin Yadav
5 years, 8 months ago

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Each way you list in Case 1 is four times more probable than each way you list in Case 2. Similar with other things. They are not equiprobable.

Ivan Koswara - 5 years, 8 months ago

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