Question

In the probability discussion post, you were presented with two options:, the post is reproduced below: o “ Suppose you're offered two chances to win money based on a simple coin flip: ▪ Option 1: If the coin lands heads, you get $5. If it lands tails, you lose $2. ▪ Option 2: If the coin lands heads, you get $15,000. If it lands tails, you lose $6,000.” • The expected value of Option 1 turned out to be +$1.50, and the expected value of Option 2 turned out to be +$4500. • Several people said that they would prefer Option 2, because the average expected result is much greater than for Option 1. However, quite a few people said they would prefer Option 1, because they were risk-averse (the prospect of losing $6000 was too hard to hear). • The standard deviation of the two possible results in option 1 is 3.5; the standard deviation of the possible results in option 2 is 10,500. • Explain what standard deviation measures, why it is associated with risk, and what it would mean for an option (like the other two above) to have a standard deviation of 0.

Answer #1

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?

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...

Suppose you toss an unfair coin 8 times independently. The
probability of getting heads is 0.3. Denote the outcome to be 1 if
you get heads and 0 if you get tails.
1.Write down the sample space.
2. What is the probability of the event that you get a head or a
tail at least once?
3. If you get 8 same toss you will get x dollars, otherwise you
will lose one dollar. On average, how large should x...

You must pick one of two wagers, for an outcome based on
flipping a fair coin.
1. You win $400 if it comes up heads and lose $100 if it comes
up tails.
2. You win $700 if it comes up heads and lose $200 if it comes
up tails.
Find the expected outcome for each wager. Which wager is better
in this sense?
The expected outcome for the first wager is
$....
(Type an integer or a decimal. Do...

Consider a game in which a coin will be flipped three times. For
each heads you will be paid $100. Assume that the coin comes up
heads with probability ⅔.
a. Construct a table of the possibilities and probabilities in
this game. The table below gives you a hint on how to do this and
shows you that there are now eight possible outcomes. (3
points)
b. Compute the expected value of the game. (2 points)
c. How much would...

1.
two coins are tossed, find the probability that two heads are
obtained. note: each coin has two possible outcomes H (heads) and T
(tails).
2. which of these numbers cannot be a probability? why?
a) -0.00001
b) 0.5
c) 20%
d)0
e) 1
3. in a deck of 52 cards, what is the probability of drawing a
three of spades, and then a four of clubs, without
replacement?
4. what is the probability of the same outcome in #3,...

Conducting a Simulation
For example, say we want to simulate the probability of getting
“heads” exactly 4 times in 10 flips of a fair coin.
One way to generate a flip of the coin is to create a vector in
R with all of the possible outcomes and then randomly select one of
those outcomes. The sample function takes a vector of elements (in
this case heads or tails) and chooses a random sample of size
elements.
coin <- c("heads","tails")...

Write a hypothesis test problem using one of the two
options below. For whichever option you choose, gather appropriate
data and post your problem (without a solution) in the discussion
topic.
Option 1:
Think about a population mean that you may be interested in and
propose a hypothesis test problem for this parameter. Gather
appropriate data and post your problem, Later, respond to your own
post with your own solution.
For example, you may believe that the population mean number...

Think about the following game: A fair coin is tossed 10 times.
Each time the toss results in heads, you receive $10; for tails,
you get nothing. What is the maximum amount you would pay to play
the game? Define a success as a toss that lands on heads. Then the
probability of a success is 0.5, and the expected number of
successes after 10 tosses is 10(0.5) = 5. Since each success pays
$10, the total amount you would...

You need to take a loan of $1,500. You have two repayment
options:
- Option 1 : Short-term 6% interest loan with a term of 1
year
- Option 2 : 1-year simple interest amortized loan at 6%
interest, monthly payments
Calculate the total interest paid for both plans.
Explain why you pay more interest with one of the options, and
which option you would prefer.

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