Question

**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 3 dollars, whereas for each tails, you lose 2 dollars. Consider the random variable

Y = the amount of money that you win.

Determine the expected value E(Y) of Y .

Answer #1

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...

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).

Alice and Bob play a game in which they flip a coin repeatedly.
Each time the coin is heads, Alice wins $1 (and Bob loses $1). Each
time the coin is tails, Bob wins (and Alice loses) $2. They
continue playing until Alice has won three flips. Prove that the
expected value of Bob’s winnings is $3. (Hint: Use linearity of
expected value to consider the expected value of each flip
separately, with flips being worth $0 if they do...

You are flipping a fair coin with one side heads, and the other
tails. You flip it 30 times.
a) What probability distribution would the above most closely
resemble?
b) If 8 out of 30 flips were heads, what is the probability of
the next flip coming up heads?
c) What is the probability that out of 30 flips, not more than
15 come up heads?
d) What is the probability that at least 15 out 30 flips are
heads?...

A player is given the choice to play this game. The player flips
a coin until they get the first Heads. Points are awarded based on
how many flips it took:
1 flip (the very first flip is Heads): 2 points
2 flips (the second flip was the first Heads): 4 points
3 flips (the third flip was the first Heads): 8 points
4 flips (the fourth flip was the first Heads): 16 points
and so on. If the player...

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) .

Suppose you flip a fair coin six times where A is the number of
tails and B is the number of heads. Prove if E(X*Y) is not equal to
E(X)E(Y).

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?

java
beginner level
NO ARRAYS in program
Flip a coin (Assignment)
How many times in a row can you flip a coin and gets heads?
Use the random number generator to simulate the flipping of the
coin. 0 means heads, 1 means tails.
Start a loop,
flip it, if heads, count it and keep flipping.
If tails, stop the loop.
Display the number of times in a row that heads came up.
Once this is working, wrap a while loop...

Suppose you flip a fair coin 10 times. What is the
probability of the last two flips both being heads if you know that
the first eight flips were heads?

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