Probability

Probability theory is a diverse field. However, when we talk about probability itself, we’re usually talking about one singular concept within the broad spectrum of topics. Probability is defined as the likelihood of something happening. The image below describes some situations in which you can use probability.probabilityAs you can see, there are many areas we can apply probability. In the second example, we can see that rain can be predicted for any given day with a specific probability. The question is, can we calculate the probability of anything? The answer is: we can calculate the probability of any event as long as it involves random variables. 

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Random Variable

Random variables are distinct from traditional variables in many ways. A random variable can be defined as a variable whose outcomes are unknown. Check out the table below to compare random variables to traditional variables.

 

Definition Notation Probability Example
Traditional Outcome is known x, y, b, a No Colour preference
Random Outcome is unknown X,Y,B,A Yes Height

 

Random variables are different from traditional, algebraic values because their outcomes cannot be determined by one solution from an equation. The outcome of a random variable can only become known in the moment that the experiment or the event actually occurs.

 

Sample Space and Event

Let’s take the classic example of a coin toss as an example of a random variable. Let’s think about what we know about the outcomes of a coin toss.

coin_flip_probability

We know that there are two outcomes: heads and tails. However, when we actually flip a coin, there is no way of knowing which outcome will occur until we check to see which side it has landed on. All possible outcomes of a random variable are defined as the sample space.

coin_sample_space

One or more outcomes in the sample space are called events. How many events are shown in the image above? The table below shows some possible answers.

 

Notation Explanation
1. E(T) Landing on tails
2. E(H) Landing on heads
3. E(T,H) First toss landing on tails, second toss heads

 

Independent Event

In the table above, we gave several examples of events for a coin toss. The equation for the probability of an event is generalized below.

probability_formula

As you can see, for the 1st and 2nd rows in the table above, the probability for either event would be 0.50, or 50%. However, what happens when we want to calculate the probability of a combination of events happening, as in the 3rd row? The first step would be to determine what type of events they are.

 

Definition
Independent The probability of one event does not depend on the outcome of another event
Mutually Exclusive (Disjoint) Two events that cannot occur at the same time
Dependent The probability of one event does depend on the outcome of another

 

Let’s take a dice roll as an example of an independent event. You’re interested in the following.

 

Notation Explanation
Sample Space S = {1,2,3,4,5,6} Can land on anything from 1 - 6
Independent Event E(1), E(2) Land on 1, then land on 2
Mutually Exclusive Event E(1 and 2) Land on 1 and  land on 2

 

The probability of each event in the second row is \frac{1}{6}. This is because landing on 1 in the first roll does not affect whether or not we will roll 6 in the next. In fact, the probability of rolling each number is the same: \frac{1}{6}.

 

The probability of landing on 1 and 2 at the same time is zero. This is an example of a mutually exclusive event because, as you can see, it is impossible to land on both 1 and 2 on one roll.

 

Dependent Event

A dependent event, as we discussed earlier, is when the probability of one event depends on the outcome of another event. A classic example of a dependent event is selection without replacement.

probability_example

In the image above, selection with replacement is illustrated as the action of selecting something from the sample and putting it back after we’re done. This would be an independent event, as the probability before and after the colour is drawn is the same.

 

Selection without replacement, on the other hand, means that the probability of the same event, which is picking red, is different before and after the selection of a card.

 

Problem 1

This question will test your knowledge on different kinds of events. You have three different sets of information: one for sample A, sample B and sample C. Given the following information of the probabilities of each event, discuss which type of event we have. Keep in mind that each sample’s events are separate.

 

Sample A Sample B Sample C
E(X) 1/3 1 1/3
E(Y) 1/3 0 2/3
E(Z) 1/3 0 0

 

Solution 1

Let’s take each sample separately.

independent_events

In the first sample, each event has the same probability. This means that we are dealing with independent events.

mutually_exclusive_events

In the second sample, we can see that if X happens then the probability of Y and Z are 0. This means these events are mutually exclusive events.

dependent_events

In the third sample, we can see each probability is different. This means that the events are dependent on each other.

 

Problem 2

There is a deck of 52 cards. There are the following cards included in the deck.

 

Queen 4
Jack 4
Ace 4
King 4
2 4
3 4
4 4
5 4
6 4
7 4
8 4
9 4
10 4

 

Find the probability of drawing a queen on the first draw and the second draw without replacement.

 

Solution 2

The image below simulates drawing 2 cards, one after the other, without replacement. First, we assume that a queen isn’t pulled the first draw.

dependent_event_example

As we can see, the probability on the first draw is 4 out of 52 cards. Because we don’t put this card back in the deck, our next draw is out of 51 cards. Also, because we haven’t drawn a queen, there are still 4 possible queens in the deck.

Next, we assume a queen is pulled on the first draw.

dependent_event_example_2

Here, because we drew a queen on the first draw, there are only 3 possible queens in the deck.

 

Problem 3

You are interested in the probability of rolling a dice. The events you are interested in are listed below.

 

P(A) Landing on 2
P(B) Landing on an even number

 

Find the sample space and probabilities for each.

 

Solution 3

Find the solution in the table below.

 

Event Sample Space Probability
P(A) Landing on 2 S = {1,2,3,4,5,6} 1/6
P(B) Landing on an even number S = {2,4,6} 3/6 = 1/2

 

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.