Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Recently I have founded some problems about probabilities, that ask to find the probability of an event at a given trial.

A dollar coin is toss several times until ones get "one dollar" face up. What is the probability to toss the coin at least $3$ times?

I thought to apply for the binomial law.But the binomial law gives the probability for a number of favorable trials, and the question ask for a specific trial.How can I solve this kind of problem?

Is there any methodology that one can apply for this kind of problems?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Hint: It is the same as the probability of two tails in a row, because you need to toss at least $3$ times precisely if the first two tosses are tails. And the probability of two tails in a row is $\frac{1}{4}$.

Remark: For fun, let's also do this problem the hard way. We need at least $6$ tosses if we toss $2$ tails then a head, or if we toss $3$ tails then a head, or we toss $4$ tails then a head, or we toss $5$ tails then a head, or $\dots$.

The probability of $2$ tails then a head is $\left(\frac{1}{2}\right)^3$. The probability of $3$ tails then a head is $\left(\frac{1}{2}\right)^4$. The probability of $4$ tails then a head is $\left(\frac{1}{2}\right)^5$. And so on. Add up. The required probability is $$\left(\frac{1}{2}\right)^3 +\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^5+\cdots.$$ This is an infinite geometric series, which can be summed in the usual way. We get $\frac{1}{4}$.

share|improve this answer
    
So, if I find the probability of getting tails in the first and second trials, I'll find the probability of the complementary event.And then will be easy to find the probability that is ask. Is that? –  João Apr 17 '12 at 17:15
    
The probability of two tails in a row is $\frac{1}{2}\times \frac{1}{2}$. That ($1/4$) is the answer to the question you asked. No complementary event needed. You need at least $3$ tosses exactly when the first two tosses are tails. –  André Nicolas Apr 17 '12 at 17:37

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.