Show work! Suppose we have the following joint probability based on historical o

Question

Show work!

Suppose we have the following joint probability based on historical observation of interest rate changes and whether or not the economy is in a recession:

In A            Not In A

Recession        Recession        |           Total

__________________________________________|______________

                                                                                                |

Interest Rates Rise            0.02                0.58            |           0.60

                                                                                    |

Interest Rates Do              0.02                0.08            |           0.10

Not Change                                                                 |

                                                                                    |

Interest Rates Fall             0.06                0.24            |           0.30

__________________________________________|______________

                                                                                    |

Total                                  0.10                0.90            |           1.00

Question 1:    What is the probability that we will be in a recession and interest rates will fall in any given year?

Question 2:    What is the probability that interest rates will fall in any given year?

Question 3:    What is the probability that we will be in a recession in any given year?

Question 4:    What is the probability that interest rates will fall given that we are in a recession?

Question 5:    What is the probability that interest rates will rise given that we are not in a recession?

Question 6:    Are interest rate changes and recessions independent events (yes or no)?

Question 7:    Support your answer to Question 52 in one sentence.

Explanation / Answer

Question 1:

Probability that we will be in a recession and interest rates will fall in any given year = 0.06 ( directly from the given table )

Question 2:

Probability that interest rates will fall in any given year= 0.30 ( directly from the given table )

Note that we used the total column here .

Question 3:

Probability that we will be in a recession in any given year = 0.1 ( directly from the given table )

Note that we used the total row here.

Question 4:

Probability that interest rates will fall given that we are in a recession is computed using bayes theorem as:

P ( interest rates fall | recession ) = P( interest rate fall and recession ) / P( recession ) = 0.06 / 0.1 = 0.6

Therefore 0.6 is the required probability here.

Question 5:

Probability that interest rates will rise given that we are not in a recession is computed using bayes theorem as:

P( rise | not in recession ) = P( rise and not in recession ) / P( not in recession ) = 0.58 / 0.9 = 0.6444

Therefore 0.6444 is the required probability here.

Question 6:

P( Interest Rates Rise) P( in recession ) = 0.6*0.1 = 0.06

but P( Interest Rates Rise and in recession ) = 0.02 which is not equal to P( Interest Rates Rise) P( in recession )

Therefore the two events are not independent here.

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