#Probability Review: Mutually Exclusive Events , Independent Events , Dependent Events , Conditional Probability , Subtraction Rule , Addition Rule , Multiplication Rule
Practice Examples : Exercise 1 Exercise 2
Exercise Questions : Solutions
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Probability Review
“In mathematics Probability is a measure of uncertainty”.
Means if we take any activity either it will be done which is counted as 100 % or 1 or it will not be done at all which is counted as 0% or 0. So, Probability of any event occurring falls between 0 and 1.
Terminology
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Probability Experiment: The experiment for which the result is uncertain is known as probability experiment. It is also called as Random experiment.
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Sample Space: The set of possible results of random experiment is called as Sample Space.
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Event: Any set of outcomes is known as Event.
Mutually Exclusive Events
If two events M and N cannot occur or impossible to occur at same time then it is known as Mutually Exclusive Events.
Two events M and N are mutually exclusive if and only if
M∩N = Ø
Probability for mutually exclusive events is;
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P (M∩N) = P(M and N) = 0
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P (M∪N) = P(M or N) = P (M) + P (N)
Example
- Drawing a card from a pack of 52 playing cards which is Queen and Jack.
- Tossing a coin once we will get either head or tail but not both
- Looking at sky and at ground.
- Driving Left and Right
Independent Events
Two events M and N will be independent if occurrence of event ‘M’ will not affect the occurrence of event ‘N’.
Probability of Independent Events;
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P (M∩N) = P(M and N) = P(M) x P(N)
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P (M∪N) = P (M) + P (N) – P(M)P(N)
Example
After tossing a coin and getting ‘Head’ in first flip is an independent event to that if you toss the coin again and get a ‘Head’.
So, the probability of getting ‘Head’ on tossing the coin both times is;
P (M) P (N) = (½ ) ( ½ )
= ¼
Note: Two events cannot be mutually exclusive and independent together. They will be either mutually exclusive or independent. Means if two events are mutually exclusive then they cannot be independent and if they are independent then they will not be mutually exclusive.
Dependent Events
Two events are dependent if the outcome or occurrence of first event M affects the outcome or occurrence of the second event N.
Conditional Probability – The probability that event M occurs multiplied by the event N occur given that event M already occurred is known as conditional probability. In conditional probability Event N is dependent on occurrence of event M. Here M and N are dependent events.
Example;
In a box there are 5 yellow and 2 green balls. Two balls are taken at random without replacement from the box. What is probability that first ball picked will be yellow and the second ball will also be yellow?
Solution
Event M = First ball taken is yellow
Event N = Second ball taken is yellow
There are total 7 balls and 5 of them are yellow. So, P (M) =
After picking first ball now there will be only 6 balls left in the box with 4 yellow and 2 green.
Probability for second ball P (N) = 4/6 = 2/3
P (M) and P (N) = 5/7 x 2/3
= 10/21
Rules and facts about Probability
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The probability of occurring an event ‘A’ is P(A) = 1
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The probability of not occurring an event ‘A’ is P(A) = 0
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The event is possible but not certain to occur, then probability
lies between 0 and 1.
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Addition Rule : The probability that event M or event N, or both occurs
P (M∪N) = P (M) + P (N) – P (M∩N)
Above rule is also known as Inclusion – Exclusion principle applied to probability.
P (M) = Probability that event M occurs
P (N) = Probability that event N occurs
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Multiplication Rule:
This rule is applied when event M and event N both occurs and are independent events. Means all outcomes in M ∩ It is the intersection of two events.
So, Probability of two events M and N both occur
P (M∩N) = P (M and N) = P (M) x P (N)
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Subtraction Rule:
The probability of an event ‘M’ will not occur is = 1 – P (M).
This rule is also known as probability of failure, i.e. the event will not occur. To recognize this type of problems always look for the word “At least”.
Example
If probability of catching a train by Rihanna from railway station is 0.85. What is the probability that Rihanna cannot get into the train?
Solution
The probability of catching the train by Rihanna, P (M) = 0.85
So, Using Subtraction rule;
Rihanna could not catch the train = 1 – P (M)
= 1.00 – 0.85
= 0.15
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Addition of the probabilities of all possible outcomes of an experiment is 1.
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The probability of an event ‘M’ is the sum of Probabilities of outcomes in M.
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Event M and N cannot be mutually exclusive and independent if P (M) ≠ 0 and P (N) ≠ 0.
Formulas
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P (M) = Probability that event M occurs
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P (N) = Probability that event N occurs
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Probability of an event ‘M’ is
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Probability for not mutually exclusive events
P (M∪N) = P (M or N) = P (M) + P (N) – P (M∩N)
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Probability for mutually exclusive events
P (M∩N) = P (M and N) = 0
P (M∪N) = P (M or N) = P (M) + P (N)
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Probability for independent events
P (M∩N) = P (M and N) = P (M) x P (N)
P (M∪N) = P (M or N) = P (M) + P (N) – P (M) P (N)
Summary
Remember mainly four equations.
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P (M and N) = P (M) P (N)
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P (M or N) = P (M) + P (N) – P (M) P (N)
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For not happening of the event = 1 – P(M), for at least one probability
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In conditional probability events M and N are dependent and the probability of event M occurs multiplied by the event N occurs given that event M already occurred.