Question

Bill pays $1 for the privilege of rolling a fair die. If an odd
number shows up, Bill will win as many dollars as the number that
shows up on the die. If an even number shows up, he will lose $3.

Find the following:

a) The probability function of Bill’s net winnings.

b) Bill’s expected net winnings and the standard deviation of
these winnings.

Answer #1

a)

below is probability mass function of Bill's net winning"

for odd numbers ; net winning =odd number-1

P(X=0)=1/6

P(X=2)=1/6

P(X=4)=1/6

P(X=-4)=P(even number)=3/6=1/2

b)

x | P(x) | xP(x) |
x^{2}P(x) |

0 | 1/6 | 0.000 | 0.000 |

2 | 1/6 | 0.333 | 0.667 |

4 | 1/6 | 0.667 | 2.667 |

-3 | 1/2 | -1.500 | 4.500 |

total | -0.500 | 7.833 | |

E(x) =μ= | ΣxP(x) = | -0.5000 | |

E(x^{2}) = |
Σx^{2}P(x) = |
7.8333 | |

Var(x)=σ^{2} = |
E(x^{2})-(E(x))^{2}= |
7.583 | |

std deviation= |
σ= √σ^{2} = |
2.7538 |

from above expected winning =-0.5

standard deviation =2.7538

A game of chance involves rolling a 15-sided die once. If a
number from 1 to 3 comes up, you win 2 dollars. If the number 4 or
5 comes up, you win 8 dollars. If any other number comes up, you
lose. If it costs 5 dollars to play, what is your expected net
winnings?

A fair die is rolled once. Let A = the die shows an odd
number. Let B = the die shows a number
greater than 4.
(a) Find A ∪ B.
(b) Find A ∩ B.
(c) Find P(A ∪ B)

A certain game consists of rolling a single fair die and pays
off as follows: $6 for a 6, $4 for a 5, $1 for a 4, and no
payoff otherwise. Find the expected winnings for this game

A fair die is rolled. Find the probability of rolling
a. a 4
b. a 5
c. a number less than 5
d. a number greater than 4
e. a number less than 20
f. a number greater than 17
g. an odd number
h. a 4 or a 5
i. a 4 or an even number
j. a 4 and a 5
k. a 4 and an even number
l. a 4 or an odd

A die is said to be fair if the probability of rolling the
number i is pi = 1/6 for i = 1, ..., 6. One student rolls a die 180
times with the following outcome.
number
1
2
3
4
5
6
frequency
38
27
33
27
28
27
What is the expected frequency for rolling 3 if the die is
fair.
Round your answer to the nearest 0.01.
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Question 21 pts
Continue the previous problem....

Consider rolling a fair 6-sided dice.
Which of the following statements are correct?
Group of answer choices
The probability that it lands on a 1 is 1/6.
The probability of an even number on one roll of a dice is
2/6.
The probability of an even number on one roll of a dice is
3/6.
If we roll this dice a large number of times, then for about 5/6
of the time, it will NOT land on a 2.
Suppose...

1) Let ? be the number that shows up when you roll a fair,
six-sided die, and, let ? = ?^2 − 5? + 6.
a. Find both formats for the distribution of ??. (Hint: tep
forms of probability distributions
are CDF and pmf/pdf.)
b.. Find F(2.35).

PROBLEM #2
Suppose you play a game in which a fair 6 sided die is rolled
once. If the outcome of the roll (the number of dots on the side
facing upward) is less than or equal to 4, you are paid as many
dollars as the number you have rolled. Otherwise, you lose as many
dollars as the number you have rolled.
Let X be the profit from the game (or the amount of money won or
lost per...

A spinner game has a wheel with the numbers 1 through 30 marked
in equally spaced slots. You pay $1 to play the game. You pick a
number from 1 to 30. If the spinner lands on your number, you win
$25. Otherwise, you win nothing. Find the expected net winnings for
this game. (Round your answer to two decimal places.)
A game costs $1 to play. A fair 5-sided die is rolled. If you
roll an even number, you...

1) A fair die is rolled 10 times. Find an expression for the
probability that at least 3 rolls of the die end up with 5 dots on
top.
2) What is the expected number of dots that show on the top of
two fair dice when they are rolled?

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