Chapter 3: Q. 3.11 (page 107)

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 $\frac{1}{2}$ ?

Short Answer

Expert verified

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

Step by step solution

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

$P (A) \geq \frac{1}{2}$

Probability of all nindependent tails has to be less than 12

Values

$P (A^{c})$

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

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

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

$P (A) \geq \frac{1}{2}$

$\Leftrightarrow$

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

$\Leftrightarrow$

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

$\Leftrightarrow$

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

$\Leftrightarrow$

$n \geq \log_{(1 - p)} \frac{1}{2}$

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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 $\frac{1}{2}$ ?

Short Answer

Step by step solution

Given value.

Probability of all nindependent tails has to be less than 12

One App. One Place for Learning.

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Study anywhere. Anytime. Across all devices.

In each of nindependent tosses of a coin, the coin lands on heads with probability p. How large need nbe so that the probability of obtaining at least one head is at least 12?

Short Answer

Step by step solution

Given value.

Probability of all nindependent tails has to be less than 12

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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 $\frac{1}{2}$ ?