Hw 6.3
A roulette wheel has 38 numbers, with 18 odd numbers (black) and 18 even numbers (red), as well as 0 and 00 (which are green). If you bet $18 that the outcome is an odd number, the probability of losing the $18 is 20/38 and the probability of winning $36 (for a net gain of only $18, given you already paid $18) is 18/38. a. If a player bets $18 that the outcome is an odd number, what is the player's expected value? b. Is the best option to bet $18 on odd or to not bet? Why?
a. -0.95 b. Not betting is best because it has the highest expected value.
If you bet $10 in a Pick 4 lottery game, you either lose $10 or gain $8,990. (The winning prize is $9,000, but your $10 bet is not returned, so the net gain is $8,990.) The game is played by selecting a four-digit number between 0000 and 9999. a. What is the probability of winning? b. If you bet $10 on 1234, what is the expected value of your gain or loss?
a. 1 / 10,000 b. -9.10
What is an expected value and how is it computed? Should we always expect to get the expected value? Why or why not?
a. Expected value is the estimated gain or loss of partaking in an event many times. EV= (event 1) x (event 1 probability) + (event 2 value) x (event 2 probability) b. We should not always expect to get the expected value because expected value is calculated with the assumption that the law of large numbers will come into play.
Suppose you toss a fair coin 100 times, getting 38 heads and 62 tails, which is 24 more tails than heads. a. Explain why, on your next toss, the difference between the numbers of heads and tails is as likely to grow to 25 as it is to shrink to 23. b. Extend your explanation from part (a) to explain why, if you toss the coin 1000 more times, the final difference between the numbers of heads and tails is as likely to be larger than 24 as it is to be smaller than 24. c. Suppose that you continue tossing the coin. Explain why the following statement is true: If you stop at any random time, you always are more likely to have fewer heads than tails, in total. d. Suppose you are betting on heads with each coin toss. After the first 100 tosses, you are well on the losing side. Explain why, if you continue to bet, you will most likely remain on the losing side. How is this answer related to the gambler's fallacy?
a. On every toss, it is just as likely to land on heads as it is to land on tails. If you toss a head, the difference becomes 23. If you toss a tail, the difference becomes 25 b. On each toss, the difference in heads and tails is equally likely to increase or decrease. After 1000 tosses, the difference is equally likely to be greater than 24 or less than 24. c. Once you have 24 more tails than heads, the difference is as likely to increase as to decrease; thus, the number of tails is likely to remain greater than the number of heads. d. Once you have fewer heads than tails, the deficit of heads is likely to remain. A streak of bad luck does not mean that a person is due for a streak of good luck.
a. What is the law of large numbers? b. Can it be applied to a single observation or experiment? Explain.
a. The law of large numbers states that if a process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). b. It does not apply to a single trial (observation or experiment), or even to small numbers of trials, but only to a large number of trials.
