1. Total number of words

Using all the letters $A,B,C,D,E$, the total number of 5-letter words is $5! = 120.$

They are arranged in dictionary order and denoted by $W_1, W_2, \dots, W_{120}$.

2. Probability pattern

Given $P(W_n)=2P(W_{n-1}), \quad n>1.$ So the probabilities form a geometric progression: $P(W_n)=2^{n-1}P(W_1).$

Since total probability is 1, $\sum_{n=1}^{120} P(W_n)=1.$ Thus, $P(W_1)(1+2+2^2+\cdots+2^{119})=1.$ Using geometric sum, $P(W_1)(2^{120}-1)=1,$ so $P(W_1)=\frac{1}{2^{120}-1}.$

Hence, $P(W_n)=\frac{2^{n-1}}{2^{120}-1}.$

3. Find the serial number of $\mathrm{CDBEA}$

We now find the rank of the word $\mathrm{CDBEA}$ in dictionary order.

Position 1: C

Words starting with $A$ come first, then words starting with $B$. For each fixed first letter, remaining letters can be arranged in $4! = 24$ ways.

So words before those starting with $C$: $2 \cdot 24 = 48.$

Position 2: D

Now consider words starting with $C$. Remaining letters are $A,B,D,E$. Among these, letters smaller than $D$ are $A,B$. For each such choice at second position, remaining 3 letters can be arranged in $3! = 6$ ways.

So additional words before $CD$: $2 \cdot 6 = 12.$

Running total: $48+12=60.$

Position 3: B

Now prefix is $CD$. Remaining letters are $A,B,E$. Letters smaller than $B$ is only $A$. For this choice, remaining 2 letters can be arranged in $2! = 2$ ways.

Additional words: $1\cdot 2=2.$

Running total: $60+2=62.$

Position 4: E

Now prefix is $CDB$. Remaining letters are $A,E$. Letter smaller than $E$ is $A$. Then remaining 1 letter can be arranged in $1! = 1$ way.

Additional words: $1.$

Running total: $62+1=63.$

Position 5: A

Now only one word remains with prefix $CDBE$, namely $CDBEA$ itself. So its rank is $63+1=64.$

Therefore, $\mathrm{CDBEA}=W_{64}.$

4. Compute its probability

Using $P(W_n)=\frac{2^{n-1}}{2^{120}-1},$ we get $P(\mathrm{CDBEA})=P(W_{64})=\frac{2^{63}}{2^{120}-1}.$

This matches the form $\frac{2^\alpha}{2^\beta-1}.$ Thus, $\alpha=63, \quad \beta=120.$ So, $\alpha+\beta=63+120=183.$

5. Comparison with stored answer

Derived answer is $183,$ which matches the stored correct answer.