Suppose you play a coin toss game in which you win $1 if a head appears and lose $1 if a tail appears. In the first 100 coin tosses, head comes up 46 times and tails comes up 54 times.
A) What percentage of times has heads come up the first 100 tosses? What is your net gain or loss at this point?
B) Suppose you toss the coin 22 more times (300 total), and at this point head has come up 47% of the time. Is this consistent with the law of large numbers? Explain. What is your net gain or loss at this point?
C) How many heads would you need in the next 100 tosses in order to break even after 400 tossed? is this likely to occur?
D) Suppose that still, behind after 400 tosses, you decide to keep playing because you are "due" for a winning streak. Explain how this belief would illustrate the gambler's fallacy.
Definition of gambler's fallacy: is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely
A) What percentage of times has heads come up the first 100 tosses? What is your net gain or loss at this point?
B) Suppose you toss the coin 22 more times (300 total), and at this point head has come up 47% of the time. Is this consistent with the law of large numbers? Explain. What is your net gain or loss at this point?
C) How many heads would you need in the next 100 tosses in order to break even after 400 tossed? is this likely to occur?
D) Suppose that still, behind after 400 tosses, you decide to keep playing because you are "due" for a winning streak. Explain how this belief would illustrate the gambler's fallacy.
Definition of gambler's fallacy: is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely
-
a) 46/100 = 46% of the times there is a head. $1.00(46) + (-$1.00)(54) = -$8.00 --> loss of $8.00
b) 0.47(300) = 141 --> gain $141 and then 0.53(300) = $159 loss --> overall a loss of $18
c) You have lost a total of $36 so you would to make this up and get (1/2) of the remaining 64 tosses
36 + (1/2)64 = 36 + 32 = 68 heads there is not a good chance that this would happen
b) 0.47(300) = 141 --> gain $141 and then 0.53(300) = $159 loss --> overall a loss of $18
c) You have lost a total of $36 so you would to make this up and get (1/2) of the remaining 64 tosses
36 + (1/2)64 = 36 + 32 = 68 heads there is not a good chance that this would happen
-
a) it was heads 46% of the time; you have lost $4
b) it is consistent with the Law of Large Numbers, because its tending towards the expected probability of 50% for each possibility - its closer than 46% to 54%.
22 more times wouldn't be 300 tosses, but lets say that there are 300 tosses and 47% were heads. That means that 141 of the 300 where heads, and you've lost $159 - $141 = $18 lost.
c) You need an extra $18 to break even, so 68 of the next 100 tosses must be heads.
d) The odds are still the same. It is not any more probable that you will get heads.
b) it is consistent with the Law of Large Numbers, because its tending towards the expected probability of 50% for each possibility - its closer than 46% to 54%.
22 more times wouldn't be 300 tosses, but lets say that there are 300 tosses and 47% were heads. That means that 141 of the 300 where heads, and you've lost $159 - $141 = $18 lost.
c) You need an extra $18 to break even, so 68 of the next 100 tosses must be heads.
d) The odds are still the same. It is not any more probable that you will get heads.