Compound Lottery:

A lottery over lotteries  is called a compound lottery. Given k  simple lotteriesand probability  a_k ≥ 0 with the compound  lottery is  the  risky alternative that yields simple  lottery Lk  with probability  a, for k = 1,  k.

Example

You flip a coin. If the coin comes up head in the first trial you get Rs.2. If tail appears, the coin is  flipped again. If  it comes up head  in the second flip, you get  Rs.4,  if  tail comes  up, the  game stops and  you  get nothing. This is a compound lottery. There are two lotteries here:

In the first, there was a single consequence (you receive Rs.2) and you get the prize with probability 1. Come to the second  lottery where each consequence occurs with equal probability. Thus, compound lottery involves 1/2 chance that you will play the first lottery and get Rs.2  for sure. In another  1/2 probability, you will play the second lottery and get either Rs.4 or 0.