Forgot basic probability stuff

[EvLer]

Homework Statement

coin tossed 70 times, find exact value for probability that number of heads is between 35 and 55

Homework Equations

NA

The Attempt at a Solution

P(heads) = 1/2;
and then I am not sure...

edit: ok, maybe something like this:
(1/2)*(35/70) - (1/2)*(55/70) ?? could someone confirm/correct?

Thanks in advance.

 

[Dick]

edit: ok, maybe something like this:
(1/2)*(35/70) - (1/2)*(55/70) ?? could someone confirm/correct?

Not even close. Try 'what is the probability of exactly 35' first. Hint: how many ways can this happen? Hint: what is the probability of each 'way'? Hint: Combinatorial coefficients. Ring a bell?

 

[EvLer]

Not even close. Try 'what is the probability of exactly 35' first. Hint: how many ways can this happen?

n!/(k!(n-k)!) with n = 70, k = 35

Hint: what is the probability of each 'way'?

1/2, so you multiply the above equation by 1/2?

Hint: Combinatorial coefficients. Ring a bell?

somewhat, but do you mind giving general approach after 'what is the probability of exactly 35'? this is not too complicated of a problem, I just got "rusty" after a year.

Thank you again.

 

[Dick]

n!/(k!(n-k)!) with n = 70, k = 35

1/2, so you multiply the above equation by 1/2?

somewhat, but do you mind giving general approach after 'what is the probability of exactly 35'? this is not too complicated of a problem, I just got "rusty" after a year.

Thank you again.

You are getting there. But you don't multiply by 1/2. You need to get exactly 35 head s and 35 tails. So the odds of one such event is (1/2)^35*(1/2)^35. After you've get the answer for 35 then you should be able to do 36,37,...,55. Add them up!

 

[radou]

somewhat, but do you mind giving general approach after 'what is the probability of exactly 35'? this is not too complicated of a problem, I just got "rusty" after a year.

The coin-tossing experiment is a typical example of a discrete random variable X given with the binomial distribution, i.e. for a given number of trials, say n, and a probability p of each trial, the probability that an outcome with the probability p occurs k times is given with $$p(X=k)=\left( \begin{array}{cc} n\\ k \end{array} \right)=p^k(1-p)^{n-k}$$.

 

[HallsofIvy]

Of course, you will have to calculate
$$\left(\begin{array}{c}70 \\ i\end{array}\right)\frac{1}{2^{70}}$$
for every i from 35 to 55 and sum.

If the problem had not said "find the exact value", I would have suggested using a normal approximation.

 

[Dick]

Maybe part of the exercise is to instill a genuine fondness for normal distributions.

 

[EvLer]

well the second part actually asks to find approximate value with continuity correction. I was thinking of using Central Limit theorem? or what should I use?
ps: sorry if my answers are dumb, this is a beginning of a second probability course and these are review excersizes; something is coming back to me but it's all mixed in my head ...

 

[Dick]

These are cookbook exercises. No theorems, please. The binomial distribution you've just been working with can be approximated by a normal distribution. What are it's mean and std deviation? Look it up. So now you just can do the approximation by getting area under the normal curve (from a table or computer). Between what limits? Now look up continuity correction.