**Bayes Theorem Examples :**

Here we are going to see some example problems on bayes theorem.

If A_{1}, A_{2}, A_{3}, .............A_{n} are mutually exclusive and exhaustive events such that P(Ai) > 0, i = 1,2,3,….n and B is any event in which P(B) > 0, then

**Question 1 :**

A firm manufactures PVC pipes in three plants viz, X, Y and Z. The daily production volumes from the three firms X, Y and Z are respectively 2000 units, 3000 units and 5000 units. It is known from the past experience that 3% of the output from plant X, 4% from plant Y and 2% from plant Z are defective. A pipe is selected at random from a day’s total production,

(i) find the probability that the selected pipe is a defective one.

(ii) if the selected pipe is a defective, then what is the probability that it was produced by plant Y ?

**Solution :**

P (pipe manufactured by X)

P(X) = 2000/10000 = 2/10

P (pipe manufactured by Y)

P(Y) = 3000/10000 = 3/10

P (pipe manufactured by Z)

P(Z) = 5000/10000 = 5/10

P(Defective pipes manufactured by X)

P(D/X) = 3/100

P(Defective pipes manufactured by Y)

P(D/Y) = 4/100

P(Defective pipes manufactured by Z)

P(D/Z) = 2/100

(i) find the probability that the selected pipe is a defective one.

P(D) = P(X) ⋅ P(D/X) + P(Y) ⋅ P(D/Y) + P(Z) ⋅ P(D/Z)

= (2/10) ⋅ (3/100) + (3/10) ⋅ (4/100) + (5/10) ⋅ (2/100)

= 6/1000 + 12/1000 + 10/1000

= (6 + 12 + 10)/1000

= 28/1000

= 7/250

(ii) if the selected pipe is a defective, then what is the probability that it was produced by plant Y ?

= [P(Y)⋅P(D/Y)]/[P(X)⋅P(D/X)+P(Y)⋅P(D/Y)+ P(Z)⋅ P(D/Z)]

= (3/10) ⋅ (4/100)/[(2/10)⋅(3/100) + (3/10) ⋅ (4/100) + (5/10) ⋅ (2/100)]

= (12/1000) / (28/1000)

= 12/28

= 3/7

**Question 2 :**

The chances of A, B and C becoming manager of a certain company are 5 : 3 : 2. The probabilities that the office canteen will be improved if A, B, and C become managers are 0.4, 0.5 and 0.3 respectively. If the office canteen has been improved, what is the probability that B was appointed as the manager?

**Solution :**

P(A) = 5/(5+3+2) = 5/12

P(B) = 3/(5+3+2) = 3/12

P(C) = 2/(5+3+2) = 2/12

P(I/A) = 0.4

P(I/B) = 0.5

P(I/C) = 0.3

= [P(B)⋅P(I/B)]/[P(A)⋅P(I/A)+P(B)⋅P(I/B)+ P(C)⋅ P(I/C)]

= (3/12)(0.5) / [(5/12)(0.4) + (3/12)(0.5) + (2/12)(0.3)]

= (15/120)/[20/120 + 15/120 + 6/120]

= (15/120) / (41/120)

= 15/41

**Question 3 :**

An advertising executive is studying television viewing habits of married men and women during prime time hours. Based on the past viewing records he has determined that during prime time wives are watching television 60% of the time. It has also been determined that when the wife is watching television, 40% of the time the husband is also watching. When the wife is not watching the television, 30% of the time the husband is watching the television. Find the probability that (i) the husband is watching the television during the prime time of television (ii) if the husband is watching the television, the wife is also watching the television.

**Solution :**

P(W) = 60/100 (wives are watching)

P(H/W) = 40/100

(When the wife is watching television, the husband is also watching)

P(H/W') = 30/100

(When the wife is not watching the television, the husband is watching the television)

(i) the husband is watching the television during the prime time of television

P(H) = P(W) P(H/W) + P(W') P(H/W')

= (60/100) (40/100) + (40/100) (30/100)

= 2400/10000 + 1200/10000

= 3600/10000

= 36/100

= 9/25

(ii) if the husband is watching the television, the wife is also watching the television.

P(W/H) = P(W) P(H/W)/[P(W) P(H/W) + P(W') P(H/W')]

= (60/100)(40/100)/[(60/100)(40/100)+(40/100) (30/100)]

= (2400/10000)/((2400+1200)/10000)

= (2400/10000)/(3600/10000)

= 2400/3600

= 2/3

After having gone through the stuff given above, we hope that the students would have understood, "Bayes Theorem Examples"

Apart from the stuff given in "Bayes Theorem Examples", if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**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**

**Time and work 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**