HW 3: 1.6 (3, 4, 6, 8) 2.1 (4, 6, 10, 12)
A biased coin lands heads with probability 23. The coin is tossed three times.
a) Given that there was at least one head in the three tosses, what is the probability that there
were at least two heads?
Let = # of
HW 1: 1.1 (5, 9) 1.2 (2, 3) 1.3 (12, 13, 14, 15)
1.1 5
Suppose a deck of 52 cards is shuffled and the top two cards are dealt.
a) How many ordered pairs of cards could possibly result as outcomes?
Let = 52.
|(, )| = ()2 = (52)2
Assuming each of these pai
HW 5: 2.5 (3, 6, 8, 12) 3.1 (4, 8, 12, 14)
A deck of cards is shuffled and dealt to four players, with each receiving 13 cards. Find:
a) the probability that the first player holds all the aces;
Let = # of aces.
~ ( = 13, = 52, = 4)
4 48
( = 4) =
HW 6: 3.2 (13, 14, 16, 17, 22) 3.3 (8, 14, 24)
3.2 13
Suppose a fair die is rolled ten times. Find numerical values for the expectations of each of
the following random variables:
Let ~ (1, , = 6) and be the iid value of the roll.
1
1
() = 2(1 + ) = 2(1
HW 4: 2.2 (5, 6, 12, 13) 2.4 (5, 7, 8, 9)
Suppose you bet a dollar on red, 25 times in a row, at roulette. Each time you win a dollar
with probability 1838, lose with probability 2038. Find, approximately, the chance that
after 25 bets you have at least
HW 2: 1.4 (6, 7, 8, 10, 11, 12) 1.5 (4, 6)
Suppose two cards are dealt from a deck of 52. What is the probability that the second card is
a spade given that the first card is black?
(1 2 ) + (1 2 )
(1 )(2 |1 ) + (1 )(2 |1 )
(1 2 )
(2 |1) =
=
=
(1 )
(1 )
HW 7: 3.4 (8, 12, 14, 17, 18) 3.5 (9, 10, 15)
Craps. In this game a player throws two dice and observes the sum. A throw of 7 or 11 is an
immediate win. A throw of 2, 3, or 12 is an immediate loss. A throw of 4, 5, 6, 8, 9, or 10
becomes the players poin