Any math whizzes out there? I’ve been trying to wrap my head around a couple of things, and it’s just not wrapping.
1) The CHEO Dream Home. There’s an early bird draw in December, and a grand prize draw in January. For the sake of simplicity, let’s say there’s only one of each: a car in December, and a house in January.
16,000 tickets were sold. Each ticket’s chances of winning the early bird draw are 1 in 16,000.
The winning ticket will be returned to the barrel after the early bird draw.
Each ticket’s chances of winning the main prize draw are 1 in 16,000.
That means the winner of the early bird draw has the same chance of winning the grand prize as anybody else.
But the odds of winning both prizes are much, much lower. So shouldn’t the Early Bird winning ticket have a much, much lower chance of winning the Grand Prize?
2) If I walk 10 kilometers at 5 km/hr, would I use the same amount of energy as if I walked 5 km at 10km/hr?
3) I set my programmable thermostat for 15C during the night, 18C when my alarm goes off, back to 15C when I leave for work, and 20C in the evenings. Is it hugely inefficient to warm my house up for that 45 minutes in the morning before work? And doesn’t it take almost as much energy to heat my house from 15 to 20 degrees as it would to keep it at 20 all day?
1. I know this one. You start over every time. So the odds of that ticket winning the grand prize are exactly the same as the odds of any other ticket winning it.
2. Yes.
3. I don’t know.
Anything else you were wondering about zoom? 😉
The answers:
1) No. Since each drawing is independent, whether or not you won the first draw has no impact on the probability of winning the second draw.
If you flip a coin, the odds of the coin turning up ‘heads’ is 1 in 2. If you flip the same coin a second time the odds of turning up ‘heads’ is still 1 in 2, even though the odds of turning up two heads in two flips is 1 in 4.
2) This is actually a physiology question, as it depends on how much energy the body uses when walking at each speed. My limited knowledge of biology leads me to believe that the increase in energy is not linear with respect to speed.
3) Unless you have a very large house with poor insulation, for the amount of time you keep your house at a lower temperature, I’d have to say you are probably better to keep your current habit.
About the CHEO dream house – I agree with you both that the odds remain 1 in 16,000 for both draws, but I don’t understand something. If the odds are extremely low that the same person will win both draws, how come the first win doesn’t in some way diminish the odds of the second win?
How would it do that? It sounds like you want it to magically reach forward through time and change the odds. That would be voodoo, not math.
(That is to say, I have no idea.)
If I win the car, it is extremely unlikely that I will win the house, because the odds of winning BOTH draws are so incredibly low. So how can my odds of winning the house remain 1 in 16,000 just like everybody else’s, if I have already won the car?
Hey Zoom,
Your odds of winning both are 1 in (16,0000 * 16,0000).
That is really, really low.
But once you’ve won the car, you’ve already beat the 1/16,000. Your ticket goes back in the drum, there are 16,000 tickets in the drum, your ticket has the same chance as anyone else’s. There is no magic force field around it that makes it more or less likely to get chosen.
2. 10 km at 5kph takes 2 hours. 5km at 10kph takes 1/2 hour.
According to this page: http://bradmackay.blogspot.com/feeds/posts/default
A 3.0mph walk burns 281 calories per hour for a 155 pound person. 5kph is pretty close to 3mph. So the walk burns 562 calories.
A 6mph run burns 704 calories per hour, so 352 calories in the half hour, and then what are you doing for the extra 1.5 hours
3. Your house at 20 degrees will have a bigger difference with the outside and so lose heat much faster than it will at 15 degrees. So it is good to keep it down at 15. That said, your morning boost to 18 is probably not so terrible. But you could set it to go back to 15 before you actually leave. There’s no point in it bringing the temp from 17 to 18 just as you’re walking out the door.
Aack, I hit enter too soon. I wasn’t finished #2.
Hey Zoom,
Your odds of winning both are 1 in (16,0000 * 16,0000).
That is really, really low.
But once you’ve won the car, you’ve already beat the 1/16,000. Your ticket goes back in the drum, there are 16,000 tickets in the drum, your ticket has the same chance as anyone else’s. There is no magic force field around it that makes it more or less likely to get chosen.
2. 10 km at 5kph takes 2 hours. 5km at 10kph takes 1/2 hour.
According to this page: http://bradmackay.blogspot.com/feeds/posts/default
A 3.0mph walk burns 281 calories per hour for a 155 pound person. 5kph is pretty close to 3mph. So the walk burns 562 calories.
A 6mph run burns 704 calories per hour, so 352 calories in the half hour, and then what are you doing for the extra 1.5 hours — let’s say you were ice fishing or playing your cello – 141 calories per hour, or 221 calories for the 1.5 hours = 493 calories for the 2 hours.
The walk uses more energy.
3. Your house at 20 degrees will have a bigger difference with the outside and so lose heat much faster than it will at 15 degrees. So it is good to keep it down at 15. That said, your morning boost to 18 is probably not so terrible. But you could set it to go back to 15 before you actually leave. There’s no point in it bringing the temp from 17 to 18 just as you’re walking out the door.
David, thank you very much for that. I now understand the lottery thing (#1) and the heat thing (#3).
But about the run/walk (#2): shouldn’t the 352 and the 221 be added together, which would be 573 for the two hours, which would make the 1/2 hour run marginally better than the 2-hour walk?
This math stuff is interesting. And even better when it’s explained out. I used to be better at reasoning things out and could make simple changes to knitting patterns when I was still in grade school. Now it’s tougher.
Sorry, Zoom, you are right. I added the wrong numbers.
You did however manage to explain it in such a way that I was able to tell the wrong numbers had been added, which is the important thing.
That’s the trickiest part of math – articulating it. I had trouble even formulating the question in a coherent way. I think it’s a gift to be able to communicate math explanations.