Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset 91影视 Explanations$ Math

13th Edition 路 888 pages

ISBN: 9780134506593

Chapter 3: Q83E (page 200)

Suppose the events B1, $B_{1}, B_{2}, B_{3}$ are mutually exclusive and complementary events, such that, $P (B_{1}) = 0 . 2$ , $P (B_{2}) = 0 . 15$ and $P (B_{3}) = 0 . 65$ . Consider another event A such that $P (A) = 0 . 4$ . If A is independent $B_{1}, B_{2}, B_{3}$ , use Bayes鈥檚 Rule to show that $P (\frac{B_{1}}{A}) = P (B_{1}) = 0 . 2$

Short Answer

Expert verified

Applying the Byes鈥檚 rule I get the result $P (\frac{B_{1}}{A}) = P (B_{1}) = 0 . 2$

Step by step solution

Important formula

The Baye鈥檚 formula is

$\begin{array}{rcl} P (\frac{B_{i}}{A}) & = & \frac{P (B_{i} 鈭� A)}{P (A)} \\ = & \frac{P (B_{i}) P (\frac{A}{B_{i}})}{P (B_{1}) P (\frac{A}{B_{1}}) + P (B_{2}) P (\frac{A}{B_{2}}) + . . . + P (B_{k}) P (\frac{A}{B_{k}})} \end{array}$

Show that PB1A=P(B1)=0.2

Apply the Baye鈥檚 rule which says that there are mutually exclusive and exhaustive $B_{1}, B_{2}, B_{3}, . . ., B_{k}$ such that $P (B_{1}) + P (B_{2}) + . . . P (B_{k}) = 1$ and another event A then .

$P (\frac{B_{i}}{A}) = \frac{P (B_{i} 鈭� A)}{P (A)}$

Now,

$\begin{array}{rcl} P (B_{1}) + P (B_{2}) + P (B_{3}) & = & 0.2 + 0.15 + 0.65 \\ = & 1 \end{array}$

A does not depends upon $B_{1}, B_{2}, B_{3}$

So,

$\begin{array}{rcl} P (\frac{B_{1}}{A}) & = & \frac{P (B_{1} 鈭� A)}{P (A)} \\ = & \frac{P (B_{1}) . P (A)}{P (A)} \\ = & P (B_{1}) \\ = & 0.2 \end{array}$

Therefore, this shows that $P (\frac{B_{1}}{A}) = P (B_{1}) = 0.2$

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Short Answer

Step by step solution

Important formula

Show that PB1A=P(B1)=0.2

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91影视

Suppose the events B1, B1,B2,B3are mutually exclusive and complementary events, such that, P(B1)=0.2, P(B2)=0.15and P(B3)=0.65. Consider another event A such that P(A)=0.4. If A is independent B1,B2,B3, use Bayes鈥檚 Rule to show thatPB1A=P(B1)=0.2

Short Answer

Step by step solution

Important formula

Show that PB1A=P(B1)=0.2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.