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Found in: Page 452

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Question: Show condition ${\mathbf{\left(}}{\mathbit{i}}{\mathbf{\right)}}$ and ${\mathbf{\left(}}{\mathbit{i}}{\mathbit{i}}{\mathbf{\right)}}$ are met under Laplace's definition of probability, when outcomes are equally likely.

Condition $\left(i\right)$ and $\left(ii\right)$ are met under Laplace's definition of probability when outcomes are equally likely.

See the step by step solution

## Step 1: Given data

Let $S$ be the sample space of an experiment with a finite or countable number of outcomes. $P\left(s\right)$ Be the probability of each outcome $S$ .

Condition :

$0⩽P\left(s\right)⩽1$ For each $s\in S.$

This states that the probability of each outcome is non negative real number which is no greater than .

Condition $\left(ii\right)$ :

$\sum _{s\in S}P\left(s\right)=1$

This states that the sum of all probabilities of all possible outcome should be $1$ .

## Step 2: Concept used Laplace’s formula

Generalization of Laplace's definition in which each ${\mathbf{n}}$ outcomes is assigned a probability of $\frac{\mathbf{1}}{\mathbf{n}}$ .

## Step 3: Solve for probability

The conditions$\left(i\right)$ and $\left(ii\right)$ are satisfied when the Laplace's definition of the probability of equally likely outcomes is used and $S$ is finite.

Hence, when there are possible outcomes ${x}_{1},{x}_{2},{x}_{n}$ the two conditions are to be satisfied.

$0⩽P\left(s\right)⩽1\text{for}i=1,2,\dots \dots \dots ...n\phantom{\rule{0ex}{0ex}}\sum _{i=1}^{n}P\left({x}_{J}\right)=1$

The function from the set of all outcomes of the sample space $S$ is called the probability distribution. Conditions $\left(i\right)$ and $\left(ii\right)$ are met under Laplace's definition of probability when outcomes are equally likely.