Math question on probabilities and coin tossing?

Question

It is known that for a large number of coin tosses, the proportion of tosses which turn up heads is approximately the same as the actual probability of obtaining a head on a single toss. In other words, Number of heads/Total Tosses= P(Head on a single toss.) This proportion is non-deterministic (unpredictable) however, but by a theorom proved called the Strong Law of large Numbers, the sequence of proportions converges (become arbitrarily close) to the value for P(head on a single toss) with probability one. (ie. 100% chance) Is the value for P (head on a single toss) a rational # r an irrational #? You may use any method you want in attempting to address this question. There may not be a correct answer, just support your reasoning for your answer. I'll be indebted to anyone who can figure it out.

Answer 1

I would say that P is rational. The value for the probability of getting a heads in a single toss in an even coin will be, of course, 0.5. Even though the proabablity of getting half heads and half tails in a large number of tosses is about 100%, and thus the number of heads tossed will converge to 50%, the actual value of P, being the probability of heads on a single toss, will still be 0.5, ie rational.

The actual number of heads in a large number of tosses will converge to 50%, and thus when the probability is worked out using [number of heads]/[number of tosses], it will be irrational, but this isn't necassarily the proabability, it is the actual out come. And yes, the Strong Law of Large Numbers predicts that the ratio stated above should equal P, but it cant be used to actually work out the probability. There is always that chance that, after 10,000 tosses, we have 7000 heads and 3000 tails, leading us to a different figure if we use [number of heads]/[number of tosses] for P, but as this cannot necassarily be right, we must use P=0.5.

So, I think P is rational, and that using [number of heads]/[number of tosses] to confirm is P as irrational is wrong.

Thats my opinion

Ashley

Answer 2

I believe that the law of strong numbers states that if {Zt} is a sequence of n random variables drawn from a distribution with mean MU, then with probability one, the limit of sample averages of the Z's goes to MU as sample size n goes to infinity.

I interpret this to be saying that the probability of tossing a heads is exactly 1/2 as the the number of tosses goes to infinity. I do not see how you interpret this law to indicate that the probability of tossing heads on any toss goes to 1 as the number of tosses goes to infinity. That makes no sense at all to me.

P has to be a rational number as it is always the quotient of two integers.

Answer 3

Of course the probability of heads is a rational number because it can be expressed as the ratio of 2 integers. The number of heads is an integer and the number of tosses is an integer.

Answer 4

No matter what, the probability of a coin landing on heads is 50 percent.

Answer 5

usally with any coin toss you have a 50% chance on landing on the head or tail.

Answer 6

its 50% assuming a fair coin

Answer 7

50%

Answer 8

well.. i study actuarial mathematics "all probability" but sorry can u make ur question more clear than this.. oh i feel like stupid but i llllove probability.. plz lemme understand it better