Q. 3.11 - A First Course in Probability

Question

In each of $n$ independent tosses of a coin, the coin lands on heads with probability $p$. How large need $n$ be so that the probability of obtaining at least one head is at least $\frac{1}{2}$?

Step-by-Step Solution

Answer

Probability of all $n$ independent tails has to be less than $\frac{1}{2}$, $n\ge {\log }_{(1-p)}\frac{1}{2}$

Given value.

A coin that lands on its head with the probability of $p$

$n$ is the number of tossing

$A$-event in which at least one head was present

Find a relationship between $n$ and $p$ that is such that

 $P\left(A\right)\ge \frac{1}{2}$

Probability of all n independent tails has to be less than 1 2

Values

$P\left(A^c\right)$

$A^c$=no heads occurred $i.e.$ all tosses are independent.

$P\left(A^c\right)=(1-p{)}^n\Rightarrow P(A)=1-(1-p{)}^n$

We can see that the following set of equivalences holds true:

$P\left(A\right)\ge \frac{1}{2}$

$\iff$

$1-(1-p{)}^n\ge \frac{1}{2}$

$\iff$

$(1-p{)}^n\le \frac{1}{2}$

$\iff$

$(1-p{)}^n\le \frac{1}{2}, (1-p)<1$

$\iff$

$n\ge {\log }_{(1-p)}\frac{1}{2}$