homework_02

Homework 02

• Due: Friday, June 15th 9:00AM

• Read: axiomatics and assorted examples in chapter 2

• Problems:

  • 3
  • 8
  • 15c
  • 16abc
  • 25
  • 36

• Theoretical Problems:

  • 13

• Let:

  • [A^c] be the outcome of events not contained in the space [A]

ch2_p03

Two dice are thrown. Let [E] be the event that the sum of the dice is odd, let [F] be the event that at least one of the dice lands on [1], and let [G] be the event that the sum is [5]. Describe the events [EF, E \cup F, FG, EF^c, EFG].

• [EF]
  • The expression defines the space where the sum of the dice is [odd] and one die lands on a [1].
  • The expression implies that secend device must be [even] as only [1 + even = odd].
• [E \cup F]
  • The expression defines the space where
    • The sum of the dice is odd.
    • One of the dice lands on [1].
    • Both the proceeding bullets are true
• [FG]
  • The expression defins the space where one of the dice lands on a [1] and the sum of the dice is [5],
  • The expression implies the second dice lands on a [4].
• [EF^c]
  • The expression defines the space where the sum of the dice is [odd] and at least one of dice does not land on a [1].
• [EFG]
  • The expression defines the space where the sum of the dice is [odd] and at least one of dice land on a [1] and sum of the dice is [5]
  • The expression implies that one dice land on a [1] and the other on a [4], a space of two possible outcomes.

ch2_p08

Suppose that [A] and [B] are mutually exclusive events for which [P(A) = .3] and [P(B) = .5]. What is the probability that

• either [A] or [B] occurs?
  • [P( A \cup B) = P(A) + P(B) = .8 ]
    • Keep in mind they are mutually exclusive.
• A occurs but B does not?
  • [P( A // B) = P(A) + P(B) - P(B) = P(A) = .30]
• Both A and B occur?
  • [P( A \cap B = P(A) + P(B) - P(A \cup B) = 0 ]

ch2_p15c

If it is assumed that all [52] poker hands are [52 \choose 5] equally likely, what is the probability of being dealt two pairs? (This occurs when the cards have denominations [a, a, b, c, d,] where [a, b, c, d] are all distinct.)

• [ \frac{ {13 \choose 1}{4 \choose 2}{12 \choose 1}{4 \choose 1}} { {2 \choose 1}} {11 \choose 1}{4 \choose 1} ]

ch2_p16abc

Poker dice is played by simultaneously rolling 5 dice. Show that

• (a) [P{no_two_alike} = .0926]
  • [ \frac{ {6 \choose 1}{5 \choose 1}{4 \choose 1}{3 \choose 1}{2 \choose 1}}{ {6 \choose 5}} ]
• (b) [P{one_pair} = .4630]
• (c) [P{two_pair} = .2315]

ch2__p25

A pair of dice is rolled until a sum of either [5] or [7] appears. Find the probability that a [5] occurs first.

Hint: Let [E_n] denote the event that a [5] occurs on the first [n -1] roll and no [5] or [7] occurs on the first [n − 1] rolls. Compute [P(E_n )] and argue that [ \sum\limits_{n = 1}^{\infty}{ P(E_n )}] is the desired probability.

• Combinations that make a 5
  • 1, 4
  • 2, 3
  • 3, 2
  • 4, 1
• Combinations that maek a 7
  • 1,6
  • 2,5
  • 3,4
  • 4,3
  • 2,5
  • 6,1
• Total possible combinations on for a pair of die
  • [ {6 \choose 1}{6 \choose 1}]

ch2_p36

Two cards are chosen at random from a deck of [52] playing cards. What is the probability that they

• (a) are both aces?
  • [\frac{ {4 \choose 2}}{ {52 \choose 2}}]
• (b) have the same value?
  • [\frac{ {4 \choose 2}}{ {52 \choose 2}}]

ch2_t13

Prove that [P(EF^c) = P(E) − P(EF)]

[E = EF \cup EF^c], since [EF] and [EF^c] are mutually exclusive therefore [P(E) = P(EF) + P(EF^c)] [\Rightarrow P(EF^c) = P(E) -P(EF)]