Question

Consider the following game. You flip an unfair coin, with P(H) = 1/4 and P(T) = 3/4, 100 times. Every time you flip a heads you win $8, and every time you flip a tails you lose $3. Let X be the amount of money you win/lose during the game. Justify your answers and show all work. Compute E(X) andCompute V (X).

Answer #1

Question 3: You are
given a fair coin. You flip this coin twice; the two flips are
independent. For each heads, you win 3 dollars, whereas for each
tails, you lose 2 dollars. Consider the random variable
X = the amount of money that you
win.
– Use the definition of expected value
to determine E(X).
– Use the linearity of expectation to
determineE(X).
You flip this coin 99 times; these
flips are mutually independent. For each heads, you win...

You select a coin at random: 2/3 of the coins are unfair, 1/3 of
the coins are fair. The fair coins are equally likely to flip heads
or tails. The unfair coins flip heads 3/4 of the times, and tails
1/4 of the times. You flip the selected coin and get heads or
tails. Find (1) the probability that the selected coin is fair
given the flip is heads, (2) the probability that the selected coin
is fair given the...

Lets's say I flip a coin which a probability p of turning up
heads. The game is structured in the way that I win the game if
heads appears x times before tails has appeared y times. How can I
represent this probability in the form of a summation? In terms of
x,y and p.

You flip a coin until getting heads. Let X be the number of coin
flips.
a. What is the probability that you flip the coin at least 8
times?
b. What is the probability that you flip the coin at least 8
times given that the first, third, and fifth flips were all
tails?
c. You flip three coins. Let X be the total number of heads. You
then roll X standard dice. Let Y be the sum of those...

You flip a fair coin. If the coin lands heads, you roll a fair
six-sided die 100 times. If the coin lands tails, you roll the die
101 times. Let X be 1 if the coin lands heads and 0 if the coin
lands tails. Let Y be the total number of times that you roll a 6.
Find P (X=1|Y =15) /P (X=0|Y =15) .

4. The Casino Control Commission takes a coin from a casino it
suspects to be unfair. They flip the coin 10 times and lands on
heads 8 times.
(a) What is the probability that a fair coin would land on heads
at least 8 times?
(b) Based on your answer to the last question, do you think the
coin is unfair? Explain.
5. Every day, Janet either takes the bus or drives her car to
work. She drives her car...

Suppose you are flipping an unfair coin 10 times. Let p be the
probability of getting tails for said coin. Define X to be the
number of heads obtained.
(a.) Describe the sample space S.
(b.) Give the values x for X.
(c.) Find the likelihood of rolling exactly four heads.
(d.) Find fX(x) .

Suppose you can place a bet in the following game. You flip a
fair coin (50-50 chance it lands heads). If it lands heads, you get
4 dollars, if it lands tails, you pay 1 dollar. This is the only
bet you can make. If you don't make the bet you will neither gain
nor lose money. What is the expected utility of not placing the
bet?

Slim and Roy are flipping a unfair coin, where the coin has the
chance to land on heads 65% of the time. Every time that the coin
lands on heads, Slim gets 500 dollars from Roy, and Roy gets 500
dollars from Slim if the coin lands on tails. If the game is played
100 times (coin is flipped 100 times), what is the probability that
Slim's winnings are $20,000 more than Roy's, where 'x' is strictly
greater than 20,000...

An unfair coin is flipped 3 times, heads is 4 times as likely to
occur than tails.
x= the number of heads.
Find the probability distribution, E(x), and σ(x).

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