Powerball Powerplay - Is it worth paying the extra $1 (meaning, does it improve the expected value of the return per $ bet)?

Short answer - not really.

Long answer:

It depends on (a) the Expected Value of the Jackpot, and (b) whether you consider improvement of a negative Expected Value of the return (less negative return per $ bet, or in other words, smaller loss) worth it.

You can read about how Powerplay works at the Powerball website.

Basically, you pay an extra dollar per bet, and in return you get enhanced returns (a multiplier from 2x to 10x) on winning price tiers at 4 main + Powerball levels and below. 5 main balls + No Powerball tier doesn't change with Powerplay, nor does the Jackpot. Also, the 10x multiplier is only possible when the Jackpot is $150M or less.

To determine the Powerplay multiplier, there are either 42 balls for Jackpots > $150M or 43 balls for Jackpots <= $150M. One ball is chosen at random, with the ball counts as follows:

2x multiplier - 24 balls

3x multiplier - 13 balls

4x multiplier - 3 balls

5x multiplier - 2 balls

10x multiplier - 1 ball (only when Jackpot is <=$150M)

As a brief primer on Expected Value, the way this is calculated in this case is, first, sum up the products of the probability of each prize tier multiplied by the payout of the tier, then, secondly, divide this sum by the number of dollars bet per play (either $2 w/o Powerplay or $3 with Powerplay). If the result is less than one, there is an expected loss in the long run.

When the Expected Value of the Jackpot is about $175M, the return per unit bet is about equal for plays with or without Powerplay. Before taxes, that return is approximately -54.1%, so for every $100 bet, you are expected to lose $54.10 before taxes. Below $175M, a Powerplay bet will result in a less negative return per dollar bet. Above $175M, the Jackpot return dominates the overall Expected Value (regular ticket + Powerplay). The reason I said "Expected Value of the Jackpot" is that the calculation depends on what your share of the Pari-mutuel jackpot will be (i.e. not the nominal value), which is a strong function of the number of tickets sold. This can be calculated, with an assumption of each ticket being a Quick Pick, but it is a subject better left for another post, as the math requires some considerable background explanation.