Can we do better than the official cop-out that the “Odds of Winning” are “Variable*”? As some variables (like number of players) are unknown, we can’t calculate a number. But can we deduce a formula?
*Your odds of winning the Gold Ball Draw are one out of the total number of Gold Ball Draw Numbers issued for that draw. The odds of the winning Gold Ball Draw Number taking home the Gold Ball Jackpot are one in the total number of balls in the draw. In the first draw, the odds of the Gold Ball being drawn are 1 in 30. If the jackpot continues to carry over and all the white balls are drawn, the odds will be 1 in 1.
CLASSIC JACKPOT
The Classic Jackpot is fixed at \$5 million.
You get a set of six Classic Draw numbers from 1 – 49 for each \$3 LOTTO 6/49 play.
Six numbers are drawn from 1 – 49.
Match all six main Classic Draw Numbers on one line to win the jackpot.
GOLD BALL DRAW
Each \$3 LOTTO 6/49 play will include a unique 10-digit Gold Ball Draw Number in addition to your six Classic Draw Numbers.
One Gold Ball Draw winning number will be drawn from all selections for each draw.
Once a Gold Ball Draw winning number is drawn, a separate Gold Ball Jackpot Draw will determine the prize – the Gold Ball Jackpot OR the guaranteed \$1 million prize.
GOLD BALL JACKPOT DRAW
- The Gold Ball Jackpot starts at \$10 million and can grow to \$68 Million!
It starts with 30 balls – 29 white balls each representing the guaranteed \$1 million prize and 1 Gold Ball representing the growing jackpot.
If a white ball is drawn, the guaranteed \$1 million prize is won. That ball is removed leaving 28 white balls and 1 gold ball for the next draw, and the jackpot grows by \$2 million.
Draws continue in this manner until the Gold Ball is drawn and the jackpot is won.
Once the Gold Ball Jackpot is won, the draw resets to 30 balls and the jackpot starts back at \$10 million.
If the Gold Ball Jackpot keeps growing until no white balls remain and only the Gold Ball is left – the jackpot can reach \$68 million!
Here's my attempt.
Expected Value of each play of Lotto 6/49 with Gold Ball Draw = Expected Value from Classic Draw + $\color{red}{\text{Expected Value from Gold Ball Draw}}$ – Cost of Ticket.
$\color{red}{\text{Expected Value(Gold Ball Draw)}}$ = Probability of winning Gold Ball Jackpot + Probability of winning $1M.
Thus Expected Value each play $= \dfrac{5 \; million}{13,983,816} + {\color{red}{\dfrac{\dfrac{(\text{1 gold ball × Gold Ball Jackpot) + [number of white balls × \$1 million}] }{\text{number of white balls + 1 gold ball}}}{\text{Number of players}}}} – 3$
$= {\color{red}{\dfrac{\dfrac{\text{Gold Ball Jackpot + number of white balls} }{\text{number of white balls + 1 gold ball}}}{\text{Number of players}}}} – 2.64244$
Best Answer
There are $\binom {49}6 = \frac{49\cdot48\cdot47\cdot46\cdot45\cdot44}{1\cdot2\cdot3\cdot4\cdot5\cdot6} = 13983816$ draws possible for the regular jackpot. Each are equally likely, so your odds of winning it are $\frac 1{13983816}$, with an expected payout of $\frac{5000000}{13983816}\approx\$0.36$. This is independent of the rest of the lottery.
If for a particular drawing,
Then the probability of a particular ticket being selected for the draw is $\frac 1T$. This comes with an automatic $\$1$ million. The gold jackpot draw is for $\ge\$9$ million (the other $\$1$ million is the automatic prize). Since $30-B$ balls have already been drawn, this jackpot is $\$(69 - 2B)$ million.
So the expectation for this part of the lottery is $$\frac1T\left(1 + \frac 1B(69-2B)\right)\cdot \$1000000 = \$1000000\cdot\dfrac{69 - B}{BT}$$
Thus the overall expected gain (payout - cost) from a single ticket is $$ \$1000000\cdot\dfrac{69 - B}{BT} + \$0.36 - \$3 = \$1000000\cdot\dfrac{69 - B}{BT} - $2.64$$
To go any further, you would need to take into account how many people play each lottery, which is likely to increase as the gold ball jackpot grows. But that is not predictable by mathematics alone.