Problem 1: Consider a coin that has a probability of heads = 0.4 1.1) We flip th
ID: 2935910 • Letter: P
Question
Problem 1: Consider a coin that has a probability of heads = 0.4 1.1) We flip the coin 10 times and obtain the following sequence: HTTTHTHTTT (a) What is the frequency" of heads in the above sequence? (The "frequency" of heads is defined as the number of times that H occurs divided by the total number of trials.) (b) Is the frequency of heads in the above sequence less than, equal to, or greater than the probability that the coin comes up heads? (c) Assume that you earn $2 every time you get a head, and "lose $1 every time that you get a tail What is your average net earning per trial for the above sequence? (Hint: Add your total net earnings and divide it by the number of trials.) (d) What is the expected value of your earnings for a single coin flip? (e) Is your average earning per trial for the above sequence less than, equal to, or greater than the expected value of your earning for a single coin flip? (That is, compare your answers to parts (c) and (d).) 1.2) If the coin is flipped 10,000 times, (a) What do you expect the frequency of heads to be? Why? Will this be exactly equal to the probability of heads? Why or why not? (b) What do you expect the average earning per trial to be? Will this be exactly equal to the expected value of your earnings on a single trial? Why or why not?Explanation / Answer
1.1) a) Frequency of heads = 3/10 = 0.3
b) The fequency is less than the probability of heads
c) Average net earning for the given trial = 2x0.3 + -1x0.7
= $ -0.1
d) Expected earning = 2x0.4 - 1x0.6
= $0.2
e) Average earning per trial is less than the expected earning of a single coin toss
1.2) a) Expected frequency of heads = 0.4x1000 = 400
As the number of trials increase, the expected value will be closer to the probability. Since 1000 is a large number, expected frequency will be equal to the probability.
b) Expected average earning = (2x400 - 1x600)/100 = $0.2
This is equal to the expected value of earning on a single trial.