Probability

Probability


One of the important steps you need to make when considering the probability of two or more events occurring. Is to decide whether they are independent or related events.

Mutually Exclusive vs. Independent

It is common for people to confuse the concepts of mutually exclusive events and independent events.

Definition of a mutually exclusive event

If event A happens, then event B cannot, or vice-versa. The two events "it rained on Tuesday" and "it did not rain on Tuesday" are mutually exclusive events. When calculating the probabilities for exclusive events you add the probabilities.

Independent events
The outcome of event A, has no effect on the outcome of event B. Such as "It rained on Tuesday" and "My chair broke at work". When calculating the probabilities for independent events you multiply the probabilities. You are effectively saying what is the chance of both events happening bearing in mind that the two were unrelated.

So, if A and B are mutually exclusive, they cannot be independent. If A and B are independent, they cannot be mutually exclusive.

If the events we chose were it rained today" and "I left my umbrella at home" they are not necessarily mutually exclusive, but they are probably not independent either, because one would think that you'd be less likely to leave your umbrella at home on days when it rains.

Example of a mutually exclusive event

What happens if we want to throw 1 and 6 in any order?
This now means that we do not mind if the first die is either 1 or 6, as we are still in with a chance. But with the first die, if 1 falls uppermost, clearly It rules out the possibility of 6 being uppermost, so the two Outcomes, 1 and 6, are exclusive. One result directly affects the other. In this case, the probability of throwing 1 or 6 with the first die is the sum of the two probabilities, 1/6 + 1/6 = 1/3.

The probability of the second die being favourable is still 1/6 as the second die can only be one specific number, a 6 if the first die is 1, and vice versa.

Therefore the probability of throwing 1 and 6 in any order with two dice is 1/3 x 1/6 = 1/18. Note that we multiplied the the last two probabilities as they were independent of each other!!!

Example of an independent event

The probability of throwing a double three with two dice is the result of throwing three with the first die and three with the second die. The total possibilities are, one from six outcomes for the first event and one from six outcomes for the second, Therefore (1/6) * (1/6) = 1/36th or 2.77%.
The two events are independent, since whatever happens to the first die cannot affect the throw of the second, the probabilities are therefore multiplied, and remain 1/36th.

Mutually Exclusive: can't happen at the same time.

Examples:

• Turning left and turning right are Mutually Exclusive (you can't do both at the same time)
• Tossing a coin: Heads and Tails are Mutually Exclusive
• Cards: Kings and Aces are Mutually Exclusive

What is not Mutually Exclusive:

• Turning left and scratching your head can happen at the same time

P(A and B) = 0   impossible

But the probability of A or B is the sum of the individual probabilities:

P(A or B) = P(A) + P(B)

Example: Scoring Goals

If the probability of:

• scoring no goals (Event "A") is 20%
• scoring exactly 1 goal (Event "B") is 15%

Then:

• The probability of scoring no goals and 1 goal is 0 (Impossible)
• The probability of scoring no goals or 1 goal is 20% + 15% = 35%

Which is written:

P(A ∩ B) = 0

P(A ∪ B) = 20% + 15% = 35%

Not Mutually Exclusive

P(A or B) = P(A) + P(B) - P(A and B)

 

 

 

 

Sample Spaces and Events

An experiment is an occurrence we observe whose result is uncertain.
An experiment is an occurrence whose result is uncertain.

 An outcome is some specific aspect of the experiment that we observe.
 The sample space for the experiment is the set of all possible outcomes.
 Event
Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable.

The event E consists of all outcomes in S which are favorable.

The complement, E', of an event E is the event that E does not occur. It is the set of all outcomes not in E. 


The union, EF, of events E and F is the event that either E occurs or F occurs (or both).

The interectsion, EF, of events E and F is the event that both E and F occur.




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Statistics Tutorial: Rules of Probability

Often, we want to compute the probability of an event from the known probabilities of other events. This lesson covers some important rules that simplify those computations.

Definitions and Notation

Before discussing the rules of probability, we state the following definitions:

.Two events are mutually exclusive if they have no sample points in common.
 .The probability that Event A occurs, given that Event B has occurred, is called a conditional probability.
 .The conditional probability of A, given B, is denoted by the symbol P(A|B).
 .The probability that event A will not occur is denoted by P(A').

Rule of Subtraction

The probability of a sample point ranges from 0 to 1.
The sum of probabilities of all the sample points in a sample space equals 1.

The rule of subtraction follows directly from these properties.

Rule of Subtraction The probability that event A will occur is equal to 1 minus the probability that event A will not occur.

P(A) = 1 - P(A')

Rule of Multiplication

The rule of multiplication applies to the following situation. We have two events from the same sample space, and we want to know the probability that both events occur.

Rule of Multiplication If events A and B come from the same sample space, the probability that both A and B occur is equal to the probability the event A occurs times the probability that B occurs, given that A has occurred.

P(A ∩ B) = P(A) * P(B|A)

Example 1

An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?

Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black.

We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15


Example 2
Suppose we repeat the experiment of Example 1; but this time we select marbles with replacement. That is, we select one marble, note its color, and then replace it in the urn before making the second selection. When we select with replacement, what is the probability that both of the marbles are black?

Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, we replace the selected marble; so there are still 10 marbles in the urn, 4 of which are black. Therefore, P(B|A) = 4/10.

Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 4/25

Rule of Addition

The rule of addition applies to the following situation. We have two events from the same sample space, and we want to know the probability that either event occurs.

Rule of Addition If events A and B come from the same sample space, the probability that event A and/or event B occur is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Note: Invoking the fact that P( A ∩ B ) = P( A )P( B | A ), the Addition Rule can also be expressed as
P(A ∪ B) = P(A) + P(B) - P(A) * P( B | A )

Example 1
A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, , and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?

Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:
P(F ∪ N) = P(F) + P(N) - P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50

Example 2
A card is drawn randomly from a deck of ordinary playing cards. You win $10 if the card is a spade or an ace. What is the probability that you will win the game?

Solution: Let S = the event that the card is a spade; and let A = the event that the card is an ace. We know the following:
There are 52 cards in the deck.
There are 13 spades, so P(S) = 13/52.
There are 4 aces, so P(A) = 4/52.
There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.

Therefore, based on the rule of addition:
P(S ∪ A) = P(S) + P(A) - P(S ∩ A)
P(S ∪ A) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13