a) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a…

Question

a) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

ScanSolve AI Tutor · Accepted Answer

Here's the breakdown of the probability questions:

a) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

a) ii) The Three Axioms of Probability (Kolmogorov Axioms) are:

Non-negativity: For any event $E$ , the probability of $E$ is non-negative. $P(E) \ge 0$
Normalization: The probability of the sample space $S$ (the set of all possible outcomes) is 1. $P(S) = 1$
Additivity (for mutually exclusive events): For any sequence of mutually exclusive events $E_1, E_2, E_3, \dots$ , the probability of their union is the sum of their individual probabilities. $P(E_1 \cup E_2 \cup E_3 \cup \dots) = P(E_1) + P(E_2) + P(E_3) + \dots$

b) i)

Determine the total number of presidents and military personnel.
Total presidents = 14
Military personnel = 8
Calculate the number of civilian presidents.
Civilian presidents = Total presidents - Military personnel
Civilian presidents = $14 - 8 = 6$
Calculate the probability of randomly selecting a civilian president.
$P(\text{civilian}) = \frac{\text{Number of civilian presidents}}{\text{Total number of presidents}}$ $P(\text{civilian}) = \frac{6}{14} = \frac{3}{7}$
The probability of selecting a president who was a civilian is $\boxed{\frac{3}{7}}$ .

b) ii)
- Experiment: A procedure that can be repeated, has a well-defined set of possible outcomes, and whose outcome is not known in advance.
- Sample Space: The set of all possible outcomes of a random experiment. It is usually denoted by $S$ .
- Event Space: A collection of possible outcomes of an experiment. It is a subset of the sample space.
- Sample Point: A single possible outcome of a random experiment. It is an element of the sample space.
- Trials: A single performance of an experiment.
c)
When a fair coin is tossed twice, the sample space $S$ is:
$S = \{HH, HT, TH, TT\}$
The total number of possible outcomes is 4.

c) i) at least one head show up
Identify outcomes with at least one head.
These outcomes are $\{HH, HT, TH\}$ .
Number of favorable outcomes = 3.
Calculate the probability.
$P(\text{at least one head}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{4}$
The probability of at least one head is $\boxed{\frac{3}{4}}$ .

c) ii) only tails show up
Identify outcomes with only tails.
This outcome is $\{TT\}$ .
Number of favorable outcomes = 1.
Calculate the probability.
$P(\text{only tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{4}$
The probability of only tails is $\boxed{\frac{1}{4}}$ .

c) iii) different side show up
Identify outcomes where different sides show up (one head and one tail).
These outcomes are $\{HT, TH\}$ .
Number of favorable outcomes = 2.
Calculate the probability.
$P(\text{different sides}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{4} = \frac{1}{2}$
The probability of different sides showing up is $\boxed{\frac{1}{2}}$ .

A) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Step-by-step solution

Determine the total number of presidents and military personnel.

Calculate the number of civilian presidents.

Calculate the probability of randomly selecting a civilian president.

Identify outcomes with at least one head.

Calculate the probability.

Identify outcomes with only tails.

Calculate the probability.

Identify outcomes where different sides show up (one head and one tail).

Calculate the probability.

Still stuck on this one?

More Mathematics Questions