7C the law of large numbers

Explain the meaning of the law of large numbers. Does this law say anything about what will happen in a single observation or​ experiment? Why or why​ not?

As the experiment is done more and more​ times, the proportion of times that a certain outcome occurs should get closer to the theoretical probability that that outcome would occur. This law does not say anything about what will happen in a single observation or experiment. Large numbers of events may show some​ pattern, but the individual events are unpredictable.

What is an expected value and how is it​ computed? Should we always expect to get the expected​ value? Why or why​ not?

Expected value is the estimated gain or loss of partaking in an event many times. EV = (event 1 value) x (event 1 probability) + (event 2 value) x (event 2 probability) 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.

a roulette wheel has 38 numbers: 18 black, 18 red, and the numbers 0 and 00 in green. assume all possible outcomes (the 38 #'s) have equal probability what is the probability of getting a red number on any spin if patrons spin the wheel 100,000 times, about how many times will a red number be the outcome

P(red)= number of ways red can occur ---------------------------------- total number of outcomes = 18/38 = 0.474 law of large numbers tells us that as the game is played more, the proportion of times that the wheel shows a red number should get closer to 0.474. in 100,000 tries, the wheel should come up red close to 47.6% of the time or about 47,400

house edge

The "house" sets the odds on casino games to ensure a profit over a period of time. the expected value to the house of each individual bet

If you toss a coin four​ times, it's much more likely to land in the order HTHT than HHHH.​ (H stands for heads and T for​ tails.)

The statement does not make sense because each outcome is equally likely since the probability of any single particular outcome is​ 1/2, so each set of outcomes have the same probability of (1/2) ^4 = 1/16

I​ haven't won in my last 25 pulls on the slot​ machine, so I must be having a bad day and​ I'm sure to lose if I play again.

The statement does not make sense because the results of repeated trials do not depend on results of earlier trials.

The expected value to me of each raffle ticket I purchased is - $0.85.

The statement makes sense because a negative expected value implies​ that, averaged over many​ tickets, you should expect to lose​ $0.85 for each raffle ticket that you buy.

Explain why the probability is the same for any particular set of ten coin toss outcomes. How does this idea affect our thinking about​ streaks?

The total number of outcomes for ten coins is 2 x 2 x 2 x 2 x 2 x 2 x 2 2 x 2 x 2 = 1024​, so every individual outcome has the same probability of 1 divided by 1024. A streak of all heads would not seem surprising since a streak of all heads is just as likely as a streak of all​ tails, or as likely as any other combination of outcomes.

a company makes electronics. one out of every 50 gadgets is faulty but the company doesn't know until a customer complains. suppose the company makes a $3 profit on the sale of a gadget but suffers an $80 loss for every faulty gadget because they have to repair. can the company expect a profit in the long term?

X= profit E(X)= $3(49/50) - $80(1/50) = 147/50 - 80/50 = 67/50 = 1.34 since the expected value is positive the company can expect to make a profit

gambler's fallacy

the mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck

Suppose you toss a fair coin​ 10,000 times. Should you expect to get exactly 5000​ heads? Why or why​ not? What does the law of large numbers tell you about the results you are likely to​ get?

you​ shouldn't expect to get exactly 5000​ heads, because you cannot predict precisely how many heads will occur. The proportion of heads should approach 0.5 as the number of tosses increases.

suppose you have 5 cards, 1,2,3,4,5. the number on each card represents the number of points you receive if you draw that card. You draw one card at random, what is the expected value?

1(1/5) + 2(1/5) + 3(1/5) + 4(1/5) + 5(1/5) = 3