I want to know how to work out the odds of randomly picking specific letters for a given word from the alphabet.
- Let's call the random word: "bobdylan"
- Total letters: 8
- Total chances to randomly pick: 7 (since b appears twice)
- With each pick, the chosen letter will be removed from the pool.
- The order chosen does not matter
What I want to work out is the odds of 0, 1, 2, 3, 4, 5, 6, 7 correct letters.
What I have done is to write a program to randomly pick letters and I have found that it nearly always picks 1 – 4 correct letters in any given go with very few 0 correct answers.
Best Answer
In the case of 7 distinct letters, we can partition the alphabet into 2 groups, one with correct letters (7) and wrong letters (26-7=19).
If we choose 7 letters and want x correct letters, we need x to be from the correct letters and 7-x to be from the wrong letters.
We note that the sample space if given by the fact that we are choosing 7 letters from the alphabet of 26.
That means, for any x $\leq 7$, the probability is:
$$ \frac{\binom{7}{x} \binom{19}{7-x}}{\binom{26}{7}}$$