If m things are distributed among '$a$' men and '$b$' women, find the probability that the number of things received by men is odd ? ( given, $a$ $\leq$ $b$)
If the expression is known, then it can be shown easily by induction. But for this question how can I derive the expression.
Best Answer
The formula is here:
https://www.chegg.com/homework-help/probability-even-number-successes-n-independent-bernoulli-tr-chapter-5.1-problem-32e-solution-9780131453401-exc
Probability of even number of successes in a series of independent trials
In your notation, it's shown that the formula is:
$$\frac{1}{2}\Big[1+(1-\frac{2a}{a+b})^m\Big]$$
In their notation, your $m$ is their $n$. To convert your problem to a Bernoulli Experiment, let $a+b = c$ and let $P= \frac{a}{c}$. Then assuming that the things are distributed randomly with uniform probability across your $c$ people, that's the probability of a 'success,' i.e. the probability of distributing an item to one of the men.