**Addition theorems of probability :**

The addition theorem in the Probability is the process of determining probability that one or more events occur.

**Theorem 1 :**

For any two events A and B, the probability that either event ‘A’ or event ‘B’ occurs or both occur is

P(AuB) = P(A) + P(B) – P(AnB)

**Theorem 2 :**

For any three events A, B and C, the probability that any one of the events occurs or any two of the events occur or all the three events occur is

P(AuBuC) = P(A)+P(B)+P(C)–P(AnB)-P(BnC)-P(AnC)+P(AnBnC)

**Theorem 3 :**

For any two mutually exclusive events A and B, the probability that either A or B occurs is given by the sum of individual probabilities of A and B.

P(AuB) = P(A) + P(B)

Note : If two events A and B are mutually exclusive, AnB = Null set. Then P(AnB) = 0

**Theorem 4 :**

For any three mutually exclusive events A, B and C, the probability that the event A or B or C occurs is given by the sum of individual probabilities of A B and C.

P(AuBuC) = P(A) + P(B) + P(C)

**Problem 1 :**

A number is selected from the first 25 natural numbers. What is the probability that it would be divisible by 4 or 7 ?

**Solution :**

Let A be the event that the number selected would be divisible by 4 and B, the event that the selected number would be divisible by 7.

Then AuB denotes the event that the number would be divisible by 4 or 7.

Next we note that

A = {4, 8, 12, 16, 20, 24}

and

B = {7, 14, 21}

whereas S = {1, 2, 3, ……... 25}.

Since AnB = Null set , the two events A and B are mutually exclusive and as such we have

P(AuB) = P(A) + P(B)

P(AuB) = (6/25) + (3/25)

**P(AuB) = 9/25**

**Problem 2 :**

A number is selected from the first 30 natural numbers. What is the probability that it would be divisible by 4 or 7 ?

**Solution :**

Let A be the event that the number selected would be divisible by 4 and B, the event that the selected number would be divisible by 7.

Then AuB denotes the event that the number would be divisible by 4 or 7.

Next we note that

A = {4, 8, 12, 16, 20, 24, 28}

B = {7, 14, 21, 28}

and

AnB = {28}

whereas S = {1, 2, 3, ……... 30}.

Here, the required probability is

P(AuB) = P(A) + P(B) - P(AnB)

P(AuB) = (6/25) + (3/25) - 1/25

**P(AuB) = 8/25**

**Problem 2 :**

There are three persons A, B and C having different ages. The probability that A survives another 5 years is 0.80, B survives another 5 years is 0.60 and C survives another 5 years is 0.50. The probabilities that A and B survive another 5 years is 0.46, B and C survive another 5 years is 0.32 and A and C survive another 5 years 0.48. The probability that all these three persons survive another 5 years is 0.26. Find the probability that at least one of them survives another 5 years.

**Solution :**

From the given information, we have the following

P(A) = 0.80

P(B) = 0.60

P(C) = 0.50,

P(AnB) = 0.46

P(BnC) = 0.32

P(AnC) = 0.48

P(AnBnC) = 0.26

The probability that at least one of them survives another 5 years in given by

= P(AuBuC)

= P(A) + P(B) + P(C) – P(AnB) - P(BnC) - P(AnC) + P(AnBnC)

= 0.80 + 0.60 + 0.50 – 0.46 – 0.32 – 0.48 + 0.26

**P(AuBuC) = 0.90**

After having gone through the stuff given above, we hope that the students would have understood "Addition theorems of probability".

Apart from the stuff given above, if you want to know more about "Addition theorems of probability", please click here

Apart from "History of statistics", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**