PROBABILITY QUESTION

If I have a 41/2 % (4.5%) chance of an event happening every year from now on, at what point will it be even money that the event will occur?
Thanks.
rugcutter

I'm afraid that it will always be 4.5%

If however you're talking about it occurring or having had occurred, that's a different story.

I am sorry I didn"t make myself clear.

In how many years will it be even money that the event will occur?

You have to think about it in terms of NOT occurring. You have a 95.5% chance of the event NOT occurring in year 1. If you slip past year 1 then in year 2, you have another 95.5% chance of it NOT occurring.

0.955 * 0.955 = 0.912025

If you make it to the third year, the calculation is:

0.955 ^ 3 = 0.870983875

and so forth down the line until you goo below 0.50% in the 16th year

.955 ^ 16 = 0.47868954844692669363421067800579

But 15 Years is so close to 50% that I'll go with 15.

.955 ^ 15 = 0.50124560046798606663268133822596

I'm sure MathExtremist is about to yell at me.

Quote: rugcutter

I am sorry I didn"t make myself clear.

Don't feel bad something similar tripped up Marilyn of "Ask Marilyn" too.

Lemme see, I think you go with 95.5% chance that it will *not* happen. So on the second year [95.5% * 95.5%] there was 91.2025% chance it would not happen *either* year. Keep going with multiplying the result times 95.5% till you break 50%. Somebody will correct me if I am wrong. [edit: s2dbaker beat me to it]

Let him yell! It is exactly the information I was looking for, and I thank you!

Well, the only thing is, the math you're doing is saying that "it would have happened 50% of the time by now", but unfortunately in your case it still hasn't happened yet, and you're looking at another year of a 95.5% chance that it won't happen this year either. The math is just telling you how unlucky you have been in the past, it's not predicting the future for you, except to say that it's 95.5% that it won't happen again this year either. And, if you look at the history it's basically telling you that you've broke about even at the point where this event has a 50% chance of occurring by now. Of course, if it goes past the 50% mark, you're just really unlucky, or you're getting cheated, whichever the case may be.

Not yelling -- just pointing out that English tense structure and math often don't go well together. The phrases "something will happen" and "something will have happened" mean two different things. Your question is better asked "Starting from now, how many years would I have to look into the future in order to have a 50% chance that the event will have occurred during that timeframe?" Getting confused about where you are in time, and what future horizon you are considering, is one of the paths toward the gambler's fallacy or the belief in the maturity of chances. For example, using s2dbaker's 15 year period, the odds are roughly 50/50 that your event will occur at least once during that timeframe. But if it doesn't occur during the first 5 years, what do you know now?

A = 0 years
B = 5 years
C = 10 years
D = 15 years
E = 20 years

If you're at time A, the chance that the event will occur between A and D is 50%. If you're at B, and the event has not occurred, the chances that the event will occur between A and D is now lower than 50% (pop quiz: what is it?). If you're at B and the event *has* occurred, the chances are 100% that the event will occur between A and D because it already has. The problem is when you start looking at the span between B and D and suggest that it's still 50% even if it hasn't happened yet. It's not 50%. What *is* 50%, at that point, is the chance that the event will occur between B and E, or put another way, starting from B, that the event will have occurred by E.

If you're at B, and the event has not occurred, the chances that the event will occur between A and D is now lower than 50% (pop quiz: what is it?). If you're at B and the event *has* occurred, the chances are 100% that the event will occur between A and D because it already has.

Answering the (pop quiz). If you're at B, and the event has not occurred, the chances that the event will occur between A and D is now still 50%. The odds that it will occur between B and D is lower than 50%.

No. Tense matters. The chance *was* 50%, but it *is* no longer.

Quote: MathExtremist

No. Tense matters. The chance *was* 50%, but it *is* no longer.

Agreed! You see, the thing is, you're not at A any longer, you're now at B. So, the chances are now less then 50%. Of course, the chances from A to D are still 50%, the only problem being is that you're at B now, you're no longer at A.

Very interesting question as it seems very similar to the chances of living or dying in a given year (or say having a claim on your travel insurance). In theory your probability of dying in any given year tends to go up (see http://www.ons.gov.uk/ons/taxonomy/index.html?nscl=Interim+Life+Tables), but for mathematical interest assume that it stays steady at 4.5% (which is quite high and corresponds to a male aged 77 in UK).

This year (say aged 77) your expectancy would be 15 years. If you made it to 78, you would still expect to live 15 years. You've not been one of the 4.5% who have died.

I had an interesting coffee discussion today on people who worked until 65 seemingly lived less than people who retired earlier. One of the false statements went as follows. People tend to live, on average, 78 years. So those aged 65 only have 13 to go. Some lucky ones live into their 80s, so quite a few must die early.

Well actually the average is nearer 18. This is because 15% on people didn't make it to 65.

One of the false statements went as follows. People tend to live, on average, 78 years. So those aged 65 only have 13 to go.

I don't know why this is a "false" statement. I guess it depends on what you believe, but, as you grow older the chances of dying would go up because of the aging process. The heart weakens, the muscles and bones start deteriorating, you know, you just get old. So, the odds of dying get higher each year.

If your odds of dying stayed the same every year, then, yes, you would expect to live another 15 years. It's just that there's other factors involved that make the odds of living go down the longer you live. If they stayed the same, then you would expect the same 4.5% chance of living through the next year, but, I don't think the odds stay the same every year.

- "I'll eat when I'm hungry, and I'll drink when I'm dry. And, if the moonshine don't kill me, I'll live 'til I die." -

Quote: JyBrd0403

One of the false statements went as follows. People tend to live, on average, 78 years. So those aged 65 only have 13 to go.

I don't know why this is a "false" statement.

Look at it this way: Do you believe that someone who is 77 years old has only one year left and is certain to die in the next year? How many years does a person 81 years old have left? The fact that someone has already survived to the age of 65 or 77 or 81 has a significant impact on the likelihood that they will survive to 78.

I am reminded of the old story about the newspaper reporter who interviewed a man celebrating his 106th birthday. The reporter ended by saying that he hoped he would be able to conduct a follow-up interview on the man's 107th birthday. The old man replied, "Oh, don't worry. I'm sure I'll live another year. I checked the actuarial tables and absolutely no one dies between 106 and 107."

Quote: Doc

Look at it this way: Do you believe that someone who is 77 years old has only one year left and is certain to die in the next year?

Well, that's the problem, it depends on his physical condition at age 77 as compared to age 65. By the time he's 77, his health may be so bad that it won't be a bad bet that he will die almost certainly in the next year. The odds for this person are not the same every year.

This reminds me more or horse racing, where, if there's 9 horses in a race the odds of any particular horse winning is 9 to 1. However, because the horses are not equally as fast, or equally as strong, and they don't have the same jockey, or trainers, the "true" odds are not 9 to 1 for each horse. That's why people like horse racing because the odds makers may give a horse a 9 to 1 chance of winning, when in reality the horses "true" odds are 2 to 1. They believe they can make money from the information they have about how fast the horse is, who the trainer is, who the jockey is, etc. And, in truth, that stuff will change the odds from 9 to 1 to let's say 2 to 1, or even 100 to 1.

The attraction for a game of chance is that the odds won't change, they always stay the same. So, if you have a 50/50 game of chance, you always have a 50% chance of winning the next hand, it doesn't matter what happened in the past. That's the attraction. Nothing is going to change the odds for a game of chance, as opposed to horse racing, or how long a person will live, there's a lot of factors that will change the odds for those two.