A Note on the Probability of Winning a Lottery when the Number of Competitors is a Binomial Random Variable

Consider a model in which a consumer faces a lottery with j other people for a prize, so that the probability of winning the prize is 1/(j+1). Now let j be a random variable, determined by the binomial distribution. Specifically, let there be n potential competitors for the consumer in the lottery, each with an independent probability of ? of being a competitor. In this note, we show how the resulting expression for the expected value of 1/(j+1) using binomial probabilities can be simplified by means of the binomial theorem.