Chapter all: Probability

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The probability of some event, denoted p(e), is usually defined in such a way that p satisfies a number of rules. Equation 5. 1: approximately probability (cid:4666)(cid:4667)= (cid:1866)(cid:1873)(cid:1865)(cid:1854)(cid:1857)(cid:1870) (cid:1867)(cid:1858) (cid:1872)(cid:1870)(cid:1853)(cid:1864)(cid:1871) (cid:1867)(cid:1858) (cid:1857)(cid:1876)(cid:1868)(cid:1857)(cid:1870)(cid:1865)(cid:1857)(cid:1866)(cid:1872)= (cid:1866)(cid:1873)(cid:1865)(cid:1854)(cid:1857)(cid:1870) (cid:1867)(cid:1858) (cid:1875)(cid:1853)(cid:1877)(cid:1871) (cid:1857)(cid:1874)(cid:1857)(cid:1866)(cid:1872) (cid:1855)(cid:1853)(cid:1866) (cid:1867)(cid:1855)(cid:1855)(cid:1873)(cid:1870) (cid:1858)(cid:1870)(cid:1857)(cid:1869)(cid:1873)(cid:1857)(cid:1866)(cid:1855)(cid:1877) (cid:1867)(cid:1858) (cid:1866)(cid:1873)(cid:1865)(cid:1854)(cid:1857)(cid:1870) (cid:1867)(cid:1858) (cid:1868)(cid:1867)(cid:1871)(cid:1871)(cid:1854)(cid:1864)(cid:1857) (cid:1867)(cid:1873)(cid:1872)(cid:1855)(cid:1867)(cid:1865)(cid:1857)(cid:1871) N (e) or m = number of favorable outcomes. N (s) or n = number of total possible outcomes. These events must be mutually exclusive, meaning they are independently occurring events. Equation 5. 2: addition rule for disjoint events (cid:4666)(cid:1827) (cid:1515)(cid:1828)(cid:4667)=(cid:4666)(cid:1827)(cid:4667)+(cid:4666)(cid:1828)(cid:4667) (cid:4666)(cid:1827) (cid:1515)(cid:1828)(cid:4667) = probability of a or b occurring. Since some events are not mutually exclusive, this will require the use of the general addition rule (equation 5. 3. ) Equation 5. 3: general addition rule (cid:4666)(cid:1827) (cid:1515)(cid:1828)(cid:4667)= (cid:4666)(cid:1827)(cid:4667)+(cid:4666)(cid:1828)(cid:4667) (cid:4666)(cid:1827)(cid:1514)(cid:1828)(cid:4667) (cid:4666)(cid:1827)(cid:1514)(cid:1828)(cid:4667) = probability that both a and b could occur. The probability of an event a happening is p(a). Since all probabilities add up to 1, this means that the complement is equal to 1 p(a).

Related Questions

Can two events with nonzero probabilities be both independent and mutually exclusive? Choose the correct answer below.

A. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one.

B. No two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability o the other must be zero.

C. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal.

D. No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.

amaranthweasel550

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

Y=0 Y=1

X=0
X=1

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?