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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 2: Q. 2.3 (page 52)

$F = F E 鈭� F E^{c}$ and $E 鈭� F = E 鈭� E^{c} F$

Short Answer

Expert verified

Use associative and distributive laws.

$F = F E 鈭� F E^{c} and E 鈭� F = E 鈭� E^{c} F$

Step by step solution

01

Given Information.

$F = F E 鈭� F E^{c} and E 鈭� F = E 鈭� E^{c} F$

02

Explanation.

$\begin{array}{r} F = F E 鈭� F E^{c} & and E 鈭� F = E 鈭� E^{c} F \\ F & = F 鈭� S & A 鈯� B 鈬� A 鈭� B = A (from def. of intersection) \\ = F 鈭� (E 鈭� E^{c}) & S = E 鈭� E^{c} (from def. of complement) \\ = F E 鈭� F E^{c} & distributive law \end{array}$

This proves:

$F = F E 鈭� F E^{c}$

$E 鈭� F = E 鈭� (F E 鈭� F E^{c})$ by previously proven identity

$= (E 鈭� F E) 鈭� F E^{c}$ associative law

role="math" localid="1649044670063" $= E 鈭� F E^{c} A 鈯� B 鈬� A 鈭� B = B$ (from def. of the union)

This proves:

$E 鈭� F = E 鈭� F E^{c}$

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Most popular questions from this chapter

Prove Proposition $4.4$ by mathematical induction.

An urn contains $n$ white and $m$ black balls, where $n$ and $m$ are positive numbers.

$(a)$ If two balls are randomly withdrawn, what is the probability that they are the same color?

$(b)$ If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color?

$(c)$ Show that the probability in part $(b)$ is always larger than the one in part $(a)$ .

Consider Example $5 o$ , which is concerned with the number of runs of wins obtained when $n$ wins and $m$ losses are randomly permuted. Now consider the total number of runs鈥攖hat is, win runs plus loss runs鈥攁nd show that

P (2 k r u n s) = \frac{(\begin{matrix} n - 1 \\ k - 1 \end{matrix}) (\begin{matrix} m - 1 \\ k - 1 \end{matrix})}{(\begin{matrix} n + m \\ n \end{matrix})} P (2 k + 1 r u n s) = \frac{(\begin{matrix} n - 1 \\ k \end{matrix}) (\begin{matrix} m - 1 \\ k - 1 \end{matrix}) + (\begin{matrix} n - 1 \\ k - 1 \end{matrix}) (\begin{matrix} m - 1 \\ k \end{matrix})}{(\begin{matrix} n + m \\ n \end{matrix})}

There are $5$ hotels in a certain town. If $3$ people check

into hotels in a day, what is the probability that they each check into a different hotel? What assumptions are you making?

If $4$ married couples are arranged in a row, find the probability that no husband sits next to his wife

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