Find experimental probability

[skilmurry]

I'm tutoring Pre-Algebra, and a test question doesn't show up in the lessons leading up so I'm not sure how to approach...

"Find the experimental probability.
A scientist performed an experiment 200 times and the results only came out positive 85 of those times. What is the probability that the result of the next experiment will be positive?"

I understand that the probability can be written 85/200 after the 200th try. Do I need to multiply 85/200 by 1/2 to figure the probability of the 201st experiment being positive? Or do I just use the 85/200 because it tells me to use experimental probability to find the answer?

 

[Steven G]

No! I understand that the experimental probability is 85/200.

After doing an experiment 200 times you are assuming, based on your findings, that the probability of a positive result is 85/200. That is all. Nothing else. So the probability that the next experiment or the 350th or the 1900 experiment is positive will be 85/200 = 17/40.

 

[BigBeachBanana]

So the probability that the next experiment or the 350th or the 1900 experiment is positive will be 85/200 = 17/40.

I would have to disagree with your last statement. If we want to predict the 350th, then we would use the prior 349 experiment results to predict the outcome. That might or might not be 17/40.

 

[Steven G]

BBB,
I appreciate your comment.
I thought that based on the 200 experiments we assume that the probability of a positive result will be 17/40.
Wouldn't we say for example that the prob. that the 17th experiment was a success is 17/40?

 

[BigBeachBanana]

BBB,
I appreciate your comment.
I thought that based on the 200 experiments we assume that the probability of a positive result will be 17/40.
Wouldn't we say for example that the prob. that the 17th experiment was a success is 17/40?

Let's say you flip a fair coin (and you don't know this is a fair coin) twice and they both showed up heads. If you were to make a bet based on this experience, you'd guess the next outcome would most likely be head because you've only seen heads so far i.e. Pr(head|2 trials)= 100%.
Now, if someone else has flipped the same coin 500 times, their experience is Pr(head|500 trials)=49%. They wouldn't make a bet on head, as Pr(tail| 500 trials)=51%.
Take an extreme person who has flipped the same coin infinitely many times, their experience is Pr(head|infinite trials)=50%. This person would say, they're equally likely.
Notice that since you only flipped the coin twice, your experience will be very different than of someone who has flipped the coin infinitely many times. We use statistics (summary historical experience), to predict future outcomes (probability). Our belief in future outcomes will change based on our historical experience. This is the core concept of the Bayesian Credibility Theory.

 

[Harry_the_cat]

BBB,
I appreciate your comment.
I thought that based on the 200 experiments we assume that the probability of a positive result will be 17/40.
Wouldn't we say for example that the prob. that the 17th experiment was a success is 17/40?

If 200 experiments have been performed, then I would say that the probability that the 17th experiment was a success is either 0 or 1.

 

[skilmurry]

I really appreciate the help from everyone! It makes sense that it would be 17/40 for the reasons you’ve discussed, but also because my suggestion of multiplying it by 1/2 was taught, but not in this context. Have a great night!!

 

[Steven G]

I am still troubled by this.
You toss a coin 200 times and a of them were heads. I tell you that I was going to toss the coin 300 more times. What do you think the probability of the 315th toss will be? Won't you say a/200?

To @Harry_the_cat, I can see why you say 0 or 1. I too thought that. The issue is that you are stating what you think the chance of the 17th experiment being a success knowing the outcome (or knowing that an outcome was recorded).

 

[BigBeachBanana]

I am still troubled bytes.
You toss a coin 200 times and a of them were heads. I tell you that I was going to toss the coin 300 more times. What do you think the probability of the 315th toss will be? Won't you say a/200?

Wouldn't you say someone who tossed 500 times is more credible than someone who only tossed 200 times?
Since I trust the 500 times more than the 200 times, I would predict the 501st toss based on the 500 tosses, and not 200. Of course, you can still use the 200 tosses experience, but it wouldn't be as credible.

I can see why you say 0 or 1. I too thought that. The issue is that you are stating what you think the chance of the 17th experiment being a success knowing the outcome (or knowing that an outcome was recorded).

We use probability to predict the uncertainty of future outcomes, and not the past because we already know for certain what has happened already. There's no need to predict. Thus, for your second argument, if we tossed 200 times, we would know for certain the 17th toss was head or tail. There's no need to guess.

 

[Steven G]

Wouldn't you say someone who tossed 500 times is more credible than someone who only tossed 200 times?
Since I trust the 500 times more than the 200 times, I would predict the 501st toss based on the 500 tosses, and not 200. Of course, you can still use the 200 tosses experience, but it wouldn't be as credible.


We use probability to predict the uncertainty of future outcomes, and not the past because we already know for certain what has happened already. There's no need to predict. Thus, for your second argument, if we tossed 200 times, we would know for certain the 17th toss was head or tail. There's no need to guess.

I agree with your 2nd paragraph.

I agree with your 1st paragraph up to a point. If you were to bet on the 501st toss only knowing the result of the 1st 200 results then you would use the information from the 1st 200. That is all I am saying. We really are not disagreeing.

 

[BigBeachBanana]

I agree with your 1st paragraph up to a point. If you were to bet on the 501st toss only knowing the result of the 1st 200 results then you would use the information from the 1st 200. That is all I am saying. We really are not disagreeing.

I think the confusion is about how much data is available to predict future outcomes.
If we tossed 200 times, and the Pr(head)=17/40, then we have no reason to believe that any future tosses won't come up head with the probability of 17/40. I believe this is what you're saying. I agree.
What I am saying is, if you were to predict the 501st toss, you would already know the 500 prior tosses. We would use this instead of the 200.
And yes, we're not really disagreeing, just want to make this clear.

 

[JeffM]

I fear that we have greatly confused this student.

We conduct an experiment 200 times and record success 85 times, an estimated 42.5%.

We conclude from that evidence that if we conduct another 100 trials, the most likely result will be 42 or 43 successes.

As to trying a 201st experiment, the answer of 42.5% is meaningless. It will be either 0 or 1 success.

Now if we do repeat the experiment another 200 times and get 115 successes, why on earth would we say that if we try a third set of 200 trials, we expect 85 successes.

Once, when young but just as foolish as I am now, I tried to make sense of the arguments about the epistemology of probability. It is more pruductive to contemplate the arguments on the nature of the Trinity. I have concluded two things: probability is useful only when thinking about repeated events, and additional information changes estimates. If you are told, as I once was, that you have cancer but have a 67% chance of surviving, the information is emotionally useless. From a practical standpoint, the prudent thing to do is to assume you are going to die. An insurance company can afford to take another perspective, but they do alter their life expectancy tables in light of changing experience.