I am in possession of two coins. One is fair so that it lands heads (H) and tails (T) with equal probability while the other coin is weighted so that it always lands H. Both coins are magical: if either is flipped and lands H then a $1 bill appears in your wallet, but when it lands T nothing happens. You may only flip a coin once per period. The interest rate is i per period. You are risk-neutral and thus only concern yourself with expected values (and not variance). For simplicity, in the questions below assume you will live forever. Suppose now that I also do not know which coin is fair and which is weighted.You pick one of the two coins at random. (a) What is your willingness to pay for this coin? (b) What is your willingness to pay for "an option" to purchase the coin, where the option works as follows: you may flip the coin once and observe the outcome. Then, if you wish, you may purchase the coin from me for the amount you determined in part 4(a). (c) What is your willingness to pay for an “n-option,” which works as follows:you may flip the coin n-times and observe the outcome. Then, if you wish,you may purchase the coin from me for the amount you determined in part4(a). (d) (Difficult) Suppose now you are competing in an auction against another bidder. You select one of the coins at random. Then, each of you get to flip the coin once for free and observe the outcome. Neither of you see the result of the flip for the other. You and the other bidder will simultaneously submit sealed bids to me. I will give the coin to the highest bidder, and the highest bidder pays his bid. The low bidder pays and receives nothing. In the event of a tie a winner will be selected with 50% probability. Find the symmetric equilibrium bidding strategies of each player. [Hint: whatis the expected value of the coin given you win the auction?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
I am in possession of two coins. One is fair so that it lands heads (H) and tails (T) with equal
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