Probability

Theoretical and Mathematical Physics

TABLE OF CONTENTS :

TABLE OF CONTENTS

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Probability is the branch of mathematics that studies the numerical description of how likely it is that an event will happen. Probability covers real-life situations where it's difficult to predict whether they will happen or not because their possible outcomes occur randomly. For example, when tossing a coin, you don't know if it will land on heads or tails. Despite this, you do have valuable information that can be used to predict how likely it is for each outcome to happen.

Probability Terminology

When talking about probability, there are a few concepts that you will need to familiarize yourself with:

An experiment is a process that can be repeated many times, producing a set of specific outcomes, ie tossing a coin or rolling a die.
An event is the outcome or set of outcomes resulting from an experiment, ie when tossing a coin, a possible event will be getting tails; when rolling a die, an event will be getting a 4.
The sample space is the set of all possible outcomes, ie the sample space when tossing a coin is heads and tails, and the sample space when rolling a die is 1, 2, 3, 4, 5, and 6.
The value that describes the probability of an event can range from 0 (zero) to 1.
A probability of 0 (zero) is considered an impossible event.
A probability of 1 refers to a certain event.
If the probability of an event is 0.5, then the event is equally likely to happen as it is not to happen.
Any event with a probability between 0 and 0.5 is considered unlikely to happen, and any event with a probability between 0.5 and 1 is considered likely to happen.

Please refer to the Events (Probability) article to learn more about the probability of different types of events.

Probability Formula

The formula to calculate the probability of an event is as follows:

$P r o b a b i l i t y o f a n y e v e n t = \frac{N u m b e r o f o u t c o m e s t h a t s a t i s f y a r e q u i r e m e n t}{T o t a l n u m b e r o f p o s s i b l e o u t c o m e s}$

Probabilities can be expressed in Fractions, decimals or percentages. For example, when tossing a coin, the probability of it landing on tails is $\frac{1}{2}$ , which is the same as saying 0.5 or 50%.

Probability Rules

The main rules of probability that you need to keep in mind when calculating probabilities are:

The probability of an event happening ranges between 0 (zero) and 1:

$0 \leq P (A) \leq 1$

The sum of the probabilities of all possible outcomes equals 1.

Complement rule: The probability that an event does not happen equals 1 minus the probability of the event happening:

$P (n o t A) = 1 - P (A)$

General addition rule: The probability of A or B happening equals the probability of A plus the probability of B minus the probability of A and B happening together:

$P (A o r B) = P (A) + P (B) - P (A a n d B)$

Addition rule for mutually exclusive events: Mutually exclusive events cannot happen at the same time. To calculate the probability of A or B happening in this case, we use the addition rule:

$P (A o r B) = P (A) + P (B)$

The rule changes because for mutually exclusive events, $P (A a n d B) = 0$

Multiplication rule for independent events: Two events are independent when the occurrence of one event does not affect the probability of another one happening. If A and B are independent, the probability of A and B happening together equals the probability of A times the probability of B:

$P (A a n d B) = P (A) \times P (B)$

Conditional Probability: In this case, the probability of an event will be affected by another event that has happened. The Conditional Probability of B given that A happened is:

$P (B | A) = \frac{P (A a n d B)}{P (A)}$

The denominator will be the probability of the given event.

How do you represent probability?

When solving probability problems, it can get very confusing to work out all the possible outcomes from an experiment. To make this task easier, you can use diagrams specially designed for this purpose to create visual representations of all the possible outcomes. These diagrams are the Venn diagram and the Tree diagram. Let's see when you need to use each one.

Venn diagrams

Venn diagrams are very useful when solving probability problems, as they help you represent events graphically. A rectangle is used to represent the sample space (S), and inside the rectangle, you can draw oval shapes representing each event. You can also include the frequencies or the probabilities of each event in the diagram. Let's see the most common scenarios that you can represent with Venn diagrams for two events, A and B:

1. Event A and B: In this case, both A and B occur, represented by the intersection of the two ovals.

Probability Venn diagram event A and B StudySmarter Venn diagram of the event A and B, Marilú García De Taylor - StudySmarter Originals

2. Event A or B: In this case, A or B or both occur, represented by the union of the two ovals.

Probability Venn diagram event A or B StudySmarter Venn diagram of the event A or B, Marilú García De Taylor - StudySmarter Originals

3. Event not A: In this case, A does not occur, and it represents the complement of A.

Probability Venn diagram event not A StudySmarter Venn diagram of the event not A, Marilú García De Taylor - StudySmarter Originals

There are 30 students in a tutor group, 15 students are studying French, 12 are studying Spanish, and 5 are studying both languages. Draw a Venn diagram to represent this information.

A = students studying French

B = students studying Spanish

Include the frequency of the intersection first, then work out the other values around it.

There are 5 students studying both languages, which leaves you with 10 students studying only French and 7 students studying only Spanish. That means that the remaining 8 students are not studying any languages.

Probability Venn diagram example StudySmarter Venn diagram example, Marilú García De Taylor - StudySmarter Originals

Please follow the link to the Venn Diagrams article to expand your knowledge about this topic.

Tree diagrams

Tree diagrams are especially useful to represent all the possible outcomes when you have two or more events happening one after the other. To create a tree diagram, draw a branch for each outcome in an event. Each branch should point to its corresponding outcome and include the probability of occurrence of each outcome.

Let's represent the possible outcomes when tossing a coin twice:

The sample space is $S = {H, T}$ where H = head and T = tail

Probability Tree diagram tossing coin twice StudySmarter Tree diagram of the event tossing a coin twice, Marilú García De Taylor - StudySmarter Originals

If you go through each branch, all the possible outcomes are: HH, HT, TH and TT. The probability of the coin landing on H or T is $\frac{1}{2}$ every time, no matter how many times you toss the coin.

You can read the Tree Diagram article to expand your knowledge about this topic.

Calculating probability

Here are a couple of examples of how to calculate probability:

Based on the information provided by the Venn Diagram that we created in the previous section:

Probability Venn diagram example StudySmarter Venn diagram example, Marilú García De Taylor - StudySmarter Originals

A = students studying French

B = students studying Spanish

Calculate the probability that a student selected at random:

a) Studies French

b) Studies Spanish

c) Studies Spanish but not French

d) Does not study any languages

Solutions:

a) $P (s t u d i e s F r e n c h) = \frac{15}{30} = \frac{1}{2}$

b) $P (s t u d i e s S p a n i s h) = \frac{12}{30} = \frac{2}{5}$

c) $P (s t u d i e s S p a n i s h b u t n o t F r e n c h) = \frac{7}{30}$

d) $P (n o t S p a n i s h o r F r e n c h) = \frac{8}{30} = \frac{4}{15}$

Based on the Tree diagram from the previous section, if you want to calculate the probability of getting two heads or two tails (HH or TT), you can proceed as follows:

1. Find the probability of getting two heads (HH). To do this, you need to multiply the probabilities along that branch. $P (H H) = P (H a n d H) = P (H) \times P (H)$

Probability Tree diagram tossing coin twice StudySmarter Tree diagram of the event tossing a coin twice, Marilú García De Taylor - StudySmarter Originals

2. Now, find the probability of getting two tails (TT).

$P (T T) = P (T a n d T) = P (T) \times P (T)$

Probability Tree diagram tossing coin twice StudySmarter Tree diagram of the event tossing a coin twice, Marilú García De Taylor - StudySmarter Originals

3. To find the probability of HH or TT happening, you need to add their probabilities together.

$P (H H o r T T) = P (H H) + P (T T)$

$P (H H o r T T) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

For more examples, check out the Probability Calculations article.

What is conditional probability?

As mentioned in the Probability Rules, conditional probability refers to the probability of an event happening, given that another event has happened. The conditional probability of B given that A happened is:

$P (B | A) = \frac{P (A a n d B)}{P (A)}$

The denominator will be the probability of the given event.

In a group of students, if one is selected at random, the probability of them liking football is 60%. The probability that the selected student likes football and is a male is 40%. If a student who likes football is selected, what is the probability that the student is also male?

F = student likes football.

M = student is male

$P (F) = 0.6$

$P (F a n d M) = 0.4$

$P (M | F) = \frac{P (F a n d M)}{P (F)}$

$P (M | F) = \frac{0.4}{0.6} = 0.667$

Please follow the link to the conditional probability article to learn more about this topic.

What is a probability distribution?

A Probability Distribution is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities. A random variable is a variable whose value is defined by the outcome of a random experiment. A variable is discrete when it can only take certain numerical values within a given interval. A variable is continuous when it can take infinite values within an interval.

Discrete Probability Distribution

A discrete Probability Distribution lists all the probabilities for each outcome of the random variable using a table.

The probability that the random variable X takes a specific value x is written like this: P (X = x)

Let's consider the experiment of tossing a coin two times. The possible outcomes are: HH, HT, TH, and TT. If we say that the variable X = Number of tails, we can see from the possible outcomes that the possible values of X are 0, 1 and 2.

Now you can represent the information above as a table:

x	0	1	2
P (X = x)	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{1}{4}$

The sum of the probabilities of all the possible outcomes equals 1: ∑ P (X = x) = 1 .

In this example: $\frac{1}{4} + \frac{2}{4} + \frac{1}{4} = \frac{4}{4} = 1$ ✔

Continuous probability distribution

The probability distribution of a continuous random variable is represented by an equation. This equation is called the probability density function, which has the following characteristics:

$y = f (x)$
$y \geq 0$ for all values of x
The area under the curve of the function is equal to 1.

The Probability Distribution study set has more about this topic!

Probability - key takeaways

Probability is the branch of mathematics that studies the numerical description of how likely it is that an event will happen.
Probability covers real-life situations that are difficult to predict whether they will happen or not because their outcomes are random.
An experiment is a process that can be repeated many times, producing a set of specific outcomes, ie tossing a coin or rolling a die.
We can express probabilities in Fractions, decimals or percentages.
Venn diagrams and Tree diagrams are useful to represent the possible outcomes of an experiment when solving probability problems.
Conditional probability refers to the probability of an event happening, given that another event has happened.
A probability distribution is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities.

Frequently Asked Questions about Probability

Probability is the branch of mathematics that studies the numerical description of how likely it is that an event will happen.

Probability of any event = Number of outcomes that satisfy a requirement / Total number of possible outcomes

There are seven probability rules:

1. The probability of an event happening ranges between 0 (zero) and 1: 0 ≤ P(A) ≤ 1

2. The sum of the probabilities of all possible outcomes equals 1.

3. Complement rule: P(not A) = 1 - P(A)

4. General addition rule: P(A or B) = P(A) + P(B) -P(A and B)

5. Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B)

6. Multiplication rule for independent events: P(A and B) = P(A) x P(B)

7. Conditional Probability: P(B|A) = P(A and B)/P(A)

A probability distribution is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities.

To create a tree diagram, draw a branch for each outcome in an event. Each branch should point to its corresponding outcome, and on each branch include the probability of occurrence of each outcome.

Flashcards in Probability52 Start learning

What is conditional probability?

Conditional probability is the probability of an event B occurring given that another event A has already occurred.

Are events A and B in conditional probability dependent or independent?

dependent

What methods can be used to calculate conditional probability?

using the formula, drawing a venn diagram or drawing a tree diagram

In an international school, there are 35 pupils in one particular class. 5 of them are French. 3 of the French students are boys. A student is picked at random from the class. What is the probability of that student being a boy given that the student is French?

0.599

if we had a bag of 12 sweets, with 6 lemon (L) and 6 strawberry (S)sweets at the beginning and would then pick two sweets, one after the other without replacing them, what would be P(S|S)?

0.417

if we had a bag of 12 sweets, with 6 lemon (L) and 6 strawberry (S)sweets at the beginning and would then pick two sweets, one after the other without replacing them, what would be P(L|S)?

0.583

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