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Here is a important topic for GMAT
PROBABILITY OF MUTUALLY NON EXCLUSIVE EVENTS:-
For event A and event B, IF both A and B has some outcome that is common to both of them, then such events are called as MUTUALLY NON-EXCLUSIVE EVENTS. They are also sometimes called as compatible events
Examples:-
1) Getting a prime number OR a number less than 4 when a dice is rolled.
Prime numbers that are found in a dice are :-2,3 and 5
Numbers that are less than 4 found in dice are:-1,2 and 3
As you can see 2 and 3 are numbers that is common to both events. Hence these event is non exclusive (or compatible with each other)
2) Solving a question :- When two students are given a problem to solve, then the probability that problem will be solved.
In this case, it is possible that student 1 and student 2, both solve the problem. Hence it is again a non-exclusive event.
3)Getting a Heart OR a King from a well shuffled deck of card
This is again a non exclusive event. There are 13 cards that belong to the heart suite and there are 4 kings in the deck as well. But one of the King belong to the heart suite or we can also say that one of the heart is a king also. Therefore this is also a non exclusive event.
FORMULA TO SOLVE FOR NON-EXCLUSIVE EVENTS:-
The formula to solve such non-exclusive events is very similar to the formula we use in set theory :-
In set theory
(A Union B)=A +B -(A Intersection B)
similarly for finding probability of Non-exclusive events
P(A or B)=P(A)+P(B)-P(A and B)
Now lets reinforce the concept with some problems:-
1) A lottery box contains 50 lottery tickets numbered 1 to 50. If a lottery ticket is drawn at random, what is the probability that the number drawn is a multiple of 3 or 5?
SOLUTION:-
There are 16 multiples of 3 in between 1-50 (3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48)
There are 10 multiples of 5 in between 1-50 (5,10,15,20,25,30,35,40,45,50)
As you can see that 15, 30 and 45 are common to both 3 and 5, these are non-exclusive events.
THEREFORE
P(multiple of 3 or 5)=P(multiple of 3)+P(Multiple of 5)-P(multiple of 3 AND 5)
P(Multiple of 3 or 5)=\(\frac{16}{50}+\frac{10}{50}-\frac{3}{50}\)
P(Multiple of 3 or 5)=\(\frac{26-3}{50}==>\frac{23}{50}\)
TRY THESE QUESTIONS FOR PRACTISE :-
What is the probability of getting a diamond or a queen from a well-shuffled deck of 52 cards?
The probability of Adam solving a geometry problem is 3/4 and the probability of Bob solving the same problem is 2/3. What is the probability that the problem is solved ?
Look out for more probability problems online and solve as many as you can.
PROBABILITY OF MUTUALLY NON EXCLUSIVE EVENTS:-
For event A and event B, IF both A and B has some outcome that is common to both of them, then such events are called as MUTUALLY NON-EXCLUSIVE EVENTS. They are also sometimes called as compatible events
Examples:-
1) Getting a prime number OR a number less than 4 when a dice is rolled.
Prime numbers that are found in a dice are :-2,3 and 5
Numbers that are less than 4 found in dice are:-1,2 and 3
As you can see 2 and 3 are numbers that is common to both events. Hence these event is non exclusive (or compatible with each other)
2) Solving a question :- When two students are given a problem to solve, then the probability that problem will be solved.
In this case, it is possible that student 1 and student 2, both solve the problem. Hence it is again a non-exclusive event.
3)Getting a Heart OR a King from a well shuffled deck of card
This is again a non exclusive event. There are 13 cards that belong to the heart suite and there are 4 kings in the deck as well. But one of the King belong to the heart suite or we can also say that one of the heart is a king also. Therefore this is also a non exclusive event.
FORMULA TO SOLVE FOR NON-EXCLUSIVE EVENTS:-
The formula to solve such non-exclusive events is very similar to the formula we use in set theory :-
In set theory
(A Union B)=A +B -(A Intersection B)
similarly for finding probability of Non-exclusive events
P(A or B)=P(A)+P(B)-P(A and B)
Now lets reinforce the concept with some problems:-
1) A lottery box contains 50 lottery tickets numbered 1 to 50. If a lottery ticket is drawn at random, what is the probability that the number drawn is a multiple of 3 or 5?
SOLUTION:-
There are 16 multiples of 3 in between 1-50 (3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48)
There are 10 multiples of 5 in between 1-50 (5,10,15,20,25,30,35,40,45,50)
As you can see that 15, 30 and 45 are common to both 3 and 5, these are non-exclusive events.
THEREFORE
P(multiple of 3 or 5)=P(multiple of 3)+P(Multiple of 5)-P(multiple of 3 AND 5)
P(Multiple of 3 or 5)=\(\frac{16}{50}+\frac{10}{50}-\frac{3}{50}\)
P(Multiple of 3 or 5)=\(\frac{26-3}{50}==>\frac{23}{50}\)
TRY THESE QUESTIONS FOR PRACTISE :-
What is the probability of getting a diamond or a queen from a well-shuffled deck of 52 cards?
ANSWER:- 4/13
The probability of Adam solving a geometry problem is 3/4 and the probability of Bob solving the same problem is 2/3. What is the probability that the problem is solved ?
ANSWER:-11/12
Look out for more probability problems online and solve as many as you can.
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