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Probability of events

The Probability of events ranges between 0 and 1, and it measures how likely it is that an event will happen. If the probability of an event is 0 (zero), it is considered impossible. If the probability of an event is 1, it is certain that it will happen. If the probability of an event is 0.5, then the event is equally likely to happen as it is not likely 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. Let's see this more clearly below.

Events (Probability) Probability of events StudySmarter The probability of events - StudySmarter Originals

Probabilities can be expressed in Fractions, decimals or percentages. For example, if an event has a probability of $\frac{1}{2}$ , it is the same as saying 0.5 or 50%.

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

If you have a bag with 6 red balls and 4 blue balls, and you take one ball out of the bag, what is the probability of that ball being blue?

$P (b a l l i s b l u e) = \frac{4}{10} = \frac{2}{5} = 0.4 = 40 %$

What are independent events?

Two events (A and B) are independent if the fact that A has happened does not affect the probability of B happening, and vice versa. For example, when tossing a coin twice, the outcome of the first event does not affect the probability of the second. The probability of getting heads the first time is $\frac{1}{2}$ , and the probability of getting heads the second time is also $\frac{1}{2}$ , the probability does not change no matter how many times you toss the coin. The outcome of the previous event has no effect on the following one.

Independent events probability formula

When two events are independent, you can use the following multiplication rule :

$P (A a n d B) = P (A) \times P (B)$ using set Notation: $P (A \cap B) = P (A) \times P (B)$

This rule can be read as the probability of A and B happening together equals the probability of A times the probability of B.

Given that $P (A) = 0.6$ , $P (B) = 0.5$ and $P (A a n d B) = 0.4$ . Prove that A and B are not independent events.

$P (A) x P (B) = 0.6 x 0.5 = 0.3$

$0.3 \neq 0.4$ therefore, A and B are not independent events

What are dependent events?

Two events (A and B) are dependent if the fact that A has happened affects the probability of B happening and vice versa.

If you choose two cards from a deck of cards without putting the card back after choosing, the probability of getting an ace on the first event is $\frac{4}{52}$ . However, the probability of getting an ace for the second card will change depending on what happened on the first event:

If the first card was an ace, the probability of getting another ace will be $\frac{3}{51}$ , because an ace has already been chosen, and we have one less card in the deck.
If the first card was not an ace, then the probability of getting an ace on the second event is $\frac{4}{51}$ .

Dependent events probability formula

The multiplication rule for dependent events is as follows:

$P (A a n d B) = P (A) \times P (B a f t e r A)$ using set Notation: $P (A \cap B) = P (A) x P (B | A)$

This rule can be read as the probability of A and B happening together equals the probability A times the probability of B after A occurred.

Going back to the previous example, the probability of getting two aces from a deck of cards without replacing cards is as follows:

A= getting an ace on the first event

B= getting an ace on the second event

$P (A \cap B) = P (A) \times P (B | A)$

$P (A \cap B) = \frac{4}{52} x \frac{3}{51}$

$P (A \cap B) = \frac{12}{2652} = 0.004 = 0.4 %$

What are mutually exclusive events?

Mutually exclusive events have no outcomes in common. Therefore, they cannot occur together. For example, getting heads or tails when tossing a coin are mutually exclusive events, as you cannot get both at the same time.

Using a Venn diagram, mutually exclusive events can be represented as follows:

Events (Probability) Mutually exclusive events Venn diagram StudySmarter Venn Diagram of mutually exclusive events, Marilu García De Taylor - StudySmarter Originals

You can learn more about Venn Diagrams too.

Mutually exclusive events probability formula

In the case of mutually exclusive events, you can use the following addition rule to calculate the combined probabilities:

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

This rule can be read as the probability of A or B happening equals the probability of A plus the probability of B.

In this case, the probability of A and B happening together is 0 (zero).

$P (A a n d B) = P (A \cap B) = 0$

The probability of getting heads or tails when tossing a coin is as follows:

A= coin landing on heads

B= coin landing on tails

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

$P (A o r B) = \frac{1}{2} + \frac{1}{2} = 1$

What are combined or compound events in probability?

Combined or compound events consist of two or more experiments being carried out together. When working with combined events, it is useful to visualise all the possible outcomes using a Tree Diagram.

If you have a bag with 12 balls: 6 red, 4 blue, and 2 yellow, and you take two balls out of the bag, replacing the ball each time. What is the probability of choosing a blue and a yellow ball?

Events (Probability) Combined events example StudySmarter Example of combined events, Marilu García De Taylor - StudySmarter Originals

Let's see this more clearly in a Tree Diagram:

Events (Probability) Tree diagram example StudySmarter Tree diagram example, Marilu García De Taylor - StudySmarter Originals

The fact that the ball is being put back in the bag each time means that the events are independent; therefore, we can use the multiplication rule to find the probability of both events happening together.

Looking at the tree diagram, we can see that there are two possible paths to follow:

Getting a blue ball first and a yellow ball second
Getting a yellow ball first and a blue ball second

Using the multiplication rule $P (A \cap B) = P (A) \times P (B$ ), both paths give you the same probability $\frac{8}{144}$ , as you can see in the tree diagram, and now you need to add them together to calculate the probability of either of the outcomes happening 1 or 2:

$P (1 o r 2) = \frac{8}{144} + \frac{8}{144} = \frac{16}{144} = \frac{1}{9} = 0.111 = 11.1 %$

Events (Probability) - Key takeaways

An event in probability is the outcome or set of outcomes resulting from an experiment.
The probability of events ranges between 0 and 1, and it measures how likely it is that an event will happen.
Two events (A and B) are independent if the fact that A has happened does not affect the probability of B happening, and vice versa.
Two events (A and B) are dependent if the fact that A has happened affects the probability of B happening and vice versa.
Mutually exclusive events are events that cannot occur together.
Combined or compound events consist of two or more experiments being carried out together.

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Frequently Asked Questions about Events (Probability)

What is an event in probability?

An event is the outcome or set of outcomes resulting from an experiment. An event is also known as a subset of the sample space.

What is the probability of a certain event?

The probability of a certain event is 1.

How do you calculate the probability of two events?

Tree diagrams are useful to represent all the possible outcomes. If the events are independent, use the multiplication rule P(A and B) = P(A) x P(B) to find the probability of both events happening together. If the events are dependent, then use the multiplication rule for dependent events P(A and B) = P(A) x P(B after A). If the events are mutually exclusive, then add their probabilities together P(A or B) = P(A) + P(B), to find how likely it is for either event to happen.

What are independent events in probability?

Two events (A and B) are independent, if the fact that A has happened does not affect the probability of B happening, and vice versa.

What are mutually exclusive events in probability?

Mutually exclusive events are events that have no outcomes in common, therefore they cannot occur together.

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Events (Probability)

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Probability of events