My son goes to the same high school as infamous Aussie rocker Angus Young, from AC/DC. If the school provided any of the inspiration for songs like “Highway to Hell” and “Dirty Deeds Done at a Discount”, I think my son will learn a lot there. In fact, he might even learn that gambling doesn’t pay.
On weekday mornings, I often walk my son to school and then hop a train into the city. As we’re approaching the school, we pass a club where they feature gambling. When we pass the club, I often say, “Hey, son. Wait here for a minute and I’ll go double your lunch money.” That always gets a big laugh. And when I’m done laughing, I get to enjoy my son’s disgusted expression as he explains, again, that the joke is not funny and has never been funny.
But if the joke is not, in fact, funny, then at least it got me to thinking about probability distributions. Just what are the odds that I could double the kid’s lunch money? How could I picture those odds graphically? And if I did double his money, would he spend it on healthy foods, or would he splurge on slushies and candy at the 7-11 after school? (Actually, I know the answer to that last one.)
So I did a little research and found that the slot machines, which are called poker machines or “pokies” here, are regulated. One official government site* says that poker machine venues must return at least 87 percent of the total amount that is bet each year to players.
Right away the odds look bad. If the Return-To-Player rate, or RTP, is just 87%, that means that, on average, the machines eat as much as 13% of your money. I checked with my stock broker and he confirmed that negative 13% is not considered a good rate of return. Bummer. But I still wondered, how likely is it that I could walk in on any given day and double my son’s $5?
Consider this scenario. Let’s say the pokie gives me one pull of the lever (or push of the button) for each 20-cent piece I pop in the slot. (By the way, it's hard getting used to 20-cent coins after growing up with quarters. I would like to take this opportunity to publically apologize to all the cashiers I have short changed in the last 3 months.) So I get 25 tries for $5. On each try, there are 2 possible outcomes: I win, or I lose. Let’s say that if I win, the pokie doubles my money.
So how many tries can I expect to win? That’s a perfect question for the binomial distribution. But first I need to know the probability of winning on any given try.
Well, one way to achieve an RTP of 87% is to arrange it so that 87% of the time the pokie simply returns your coin and 13% of the time it keeps your coin. But that’s not fun. We want a chance to double our money. So in order to maintain the same RTP, if you double the payoff, you must half the odds. So the odds of winning on any given try would be 43.5% or 0.435.
Now we can look at some distributions. To get the graph below, I chose Graph > Probability Distribution Plot, then I chose View Single, and clicked OK. Then I filled in the dialog box as follows:
The graph shows that the most likely outcome is that I’ll win on 11 out of 25 tries. The tallest bar, at 11 on the x-axis, is associated with a probability of just under 0.16.
But in order to break even, I’ll need to win at least 13 of the 25 trials. If only there were some way to get the probability of that. Well, there is! Just double-click on any bar on the plot and you can specify the “Shaded Area” to visualize the probability of winning on 13 or more of the tries:
Looks like the odds of winning at least 13 times out of 25 are only 0.2548. Not great.
What if I dipped into my own lunch money? How many times would I have to pull the lever to get that elusive 13th win? Wouldn’t it be nice if there was a way to visualize that? Well there is. We can use the binomial distribution’s pessimistic cousin, the negative binomial distribution.
The negative binomial distribution models how many trials it will take for a certain event to occur a certain number of times. We can use the same dialog box that we did before:
The negative binomial plot shows that in order to get 13 wins, on average it will take 28 tries. My odds of making money don’t look any better on this plot than they did on the last one. In order to make money instead of losing it, I have to get my 13th win in 25 tries or less. So what are the odds of that? You probably already know, but let’s double-click the bars and find out:
Hey, that number looks familiar. Yup, it’s the same number we got from the binomial distribution. It’s just two different ways of looking at the same thing. The odds of getting 13 wins within the first 25 tries is 0.2548. The same probability we found with the binomial plot when we looked at the distribution of wins when the number of trials is fixed at 25.
Of course I promised my son I’d double his money, so I actually need to win all 25 of the tries. Those are some pretty small odds. In fact those odds are just 0.435^25 = 0.0000000009, which is really, really, really low.
Seems like the better bet is to let my son keep his lunch money so that he can eat well, pay attention in class, stay away from rock and roll (although I hear that Rock and Roll did pretty well by Angus Young), avoid gambling, go to a good college, get a good job and make lots of money so that he can support me when I retire at 60 and circumnavigate the globe in my private yacht with full crew including captain, deck hands, and of course a gourmet chief. At least those odds are better than trying to strike it rich playing the pokies.