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Chapter 1: Q10E (page 21)

For two arbitrary events A and B, prove that

\({\bf{Pr}}\left( {\bf{A}} \right){\rm{ }} = {\rm{ }}{\bf{Pr}}({\bf{A}} \cap {\bf{B}}){\rm{ }} + {\rm{ }}{\bf{Pr}}({\bf{A}} \cap {{\bf{B}}^c}).\)

Short Answer

Expert verified

\(\Pr \left( {\mathop{\rm A}\nolimits} \right) = \Pr \left( {{\mathop{\rm A}\nolimits} \cap {{\mathop{\rm B}\nolimits} ^c}} \right) + \Pr \left( {{\mathop{\rm A}\nolimits} \cap {\mathop{\rm B}\nolimits} } \right)\)

Step by step solution

01

Given information

There are two arbitrary events, A and B. Arbitrary events mean any events not defined specifically.

02

State the simple events and compute the probability

From Theorem 1.4.11,

\(\begin{aligned}{}{\rm{n}}\left( {{\mathop{\rm A}\nolimits} \cap {{\mathop{\rm B}\nolimits} ^c}} \right) = {\rm{n}}\left( {\mathop{\rm A}\nolimits} \right) - {\rm{n}}\left( {{\mathop{\rm A}\nolimits} \cap {\mathop{\rm B}\nolimits} } \right)\\\Pr \left( {{\mathop{\rm A}\nolimits} \cap {{\mathop{\rm B}\nolimits} ^c}} \right) = \Pr \left( {\mathop{\rm A}\nolimits} \right) - {\mathop{\rm P}\nolimits} \left( {{\mathop{\rm A}\nolimits} \cap {\mathop{\rm B}\nolimits} } \right)\\\Pr \left( {{\mathop{\rm A}\nolimits} \cap {{\mathop{\rm B}\nolimits} ^c}} \right) + \Pr \left( {{\mathop{\rm A}\nolimits} \cap {\mathop{\rm B}\nolimits} } \right) = \Pr \left( {\mathop{\rm A}\nolimits} \right)\end{aligned}\)

Therefore,

\(\Pr \left( {\mathop{\rm A}\nolimits} \right) = \Pr \left( {{\mathop{\rm A}\nolimits} \cap {{\mathop{\rm B}\nolimits} ^c}} \right) + \Pr \left( {{\mathop{\rm A}\nolimits} \cap {\mathop{\rm B}\nolimits} } \right)\)

Hence, it is proved.

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Most popular questions from this chapter

Suppose that the heights of the individuals in a certain population have a normal distribution for which the value of the mean 胃 is unknown and the standard deviation is 2 inches. Suppose also that the prior distribution of 胃 is a normal distribution for which the mean is 68 inches and the standard deviation is 1 inch. Suppose finally that 10 people are selected at random from the population, and their average height is found to be 69.5 inches.

a. If the squared error loss function is used, what is the Bayes estimate of 胃?

b. If the absolute error loss function is used, what is the Bayes estimate of 胃? (See Exercise 7 of Sec. 7.3).

Suppose that one card is to be selected from a deck of 20 cards that contains 10 red cards numbered from 1 to 10 and 10 blue cards numbered from 1 to 10. Let A be the event that a card with an even number is selected, let B be the event that a blue card is selected, and let C be the event that a card with a number less than 5 is selected. Describe the sample space S and describe each of the following events both in words and as subsets of S:

a. \({\bf{A}} \cap {\bf{B}} \cap {\bf{C}}\)

b. \({\bf{B}} \cap {{\bf{C}}^{\bf{C}}}\)

c. \({\bf{A}} \cup {\bf{B}} \cup {\bf{C}}\)

d. \({\bf{A}} \cap {\bf{(B}} \cup {\bf{C)}}\)

e. \({{\bf{A}}^{\bf{c}}} \cap {{\bf{B}}^{\bf{c}}} \cap {{\bf{C}}^{\bf{c}}}\)

Question:Suppose that a point (X, Y ) is chosen at random from the rectangle S defined as follows:\(S = \left\{ {\left( {x,y} \right):0 \le x \le 2\,and\,1 \le 4} \right\}\).

a. Determine the joint pdf. of X and Y , the marginal pdf. of X, and the marginal pdf. of Y .

b. Are X and Y independent?

Suppose that four guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.

A deck of 52 cards contains four aces. If the cards are shuffled and distributed in a random manner to four players so that each player receives 13 cards, what is the probability that all four aces will be received by the same

player?
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