Section 7C

Section 7C

The Law of Large Numbers

Open Ended

1. 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?

The Law of Large Numbers

Consider an event A with probability P(A) in a single trial. The law of large ­numbers holds that:

- For a large number of trials, the proportion in which event A occurs will be close to the probability P(A).

- The larger the number of trials, the closer the proportion should be to P(A).

This law holds as long as each trial is independent of prior trials, so that an ­individual trial always has the same probability, P(A).

Open Ended

3. 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

Consider two events, each with its own value and probability. The expected value based on these two events is

expected value = 
 $\left(value\ ofevent\ 1\right)\times\left(prob\ of\ event\ 1\right)\ +\left(value\ of\ event\ 2\right)\times\left(prob\ of\ event\ 2\right)$ 

This formula can be extended to any number of events by including more terms in the sum.

On AVERAGE this is the amount you can expect to gain/lose for each turn/policy/event.

Multiple Choice

Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

7. The expected value to me of each raffle ticket I purchased

is − $0.85.

• Makes Sense

• Does not make sense

Multiple Choice

Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

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

• Makes Sense

• Does not make sense

Flip a coin 4 time. Which is more likely? HTHT or HHHH

HHHH, HHHT, HHTH, HTHH,

THHH, HHTT, HTHT, THHT,

HTTH, THTH, TTHH, HTTT,

THTT, TTHT, TTTH, TTTT

Open Ended

13. Understanding the Law of Large Numbers. 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?

Fill in the Blanks

19–20: Insurance Claims. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss? Explain.

19. An insurance policy sells for $300. Based on past data an average of 1 in 100 policyholders will file a $10,000 claim, an average of 1 in 250 policyholders will file a $25,000 claim, and an average of 1 in 500 policyholders will file a $50,000 claim.

19. An insurance policy sells for $300. Based on past data an average of 1 in 100 policyholders will file a $10,000 claim, an average of 1 in 250 policyholders will file a $25,000 claim, and an average of 1 in 500 policyholders will file a $50,000 claim.

•  $300\left(1\right)\ +\left(-10,000\right)\left(\frac{1}{100}\right)+\left(-25,000\right)\left(\frac{1}{250}\right)+\left(-50,000\right)\left(\frac{1}{500}\right)$  

•  $300+\left(-100\right)+\left(-100\right)+\left(-100\right)$  

•  $ex\rho ected\ value\ =\ 0$  

Fill in the Blanks

19–20: Insurance Claims. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss? Explain.

20. An insurance policy sells for $600. Based on past data, an average of 1 in 50 policyholders will file a $5000 claim, an average of 1 in 100 policyholders will file a $10,000 claim, and an average of 1 in 200 policyholders will file a $30,000 claim.

20. An insurance policy sells for $600. Based on past data, an average of 1 in 50 policyholders will file a $5000 claim, an average of 1 in 100 policyholders will file a $10,000 claim, and an average of 1 in 200 policyholders will file a $30,000 claim.

•  $600+\left(-5,000\right)\left(\frac{1}{50}\right)+\left(-10,000\right)\left(\frac{1}{100}\right)+\left(-30,000\right)\left(\frac{1}{200}\right)$  

•  $600+\left(-100\right)+\left(-100\right)+\left(-150\right)$  

• $ $250\ per\ policy\ on\ average$  (expected value)

Fill in the Blanks

19–20: Insurance Claims. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss? Explain.

20. An insurance policy sells for $600. Based on past data, an average of 1 in 50 policyholders will file a $5000 claim, an average of 1 in 100 policyholders will file a $10,000 claim, and an average of 1 in 200 policyholders will file a $30,000 claim.

Fill in the Blanks

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

a. What percentage of times has heads come up in the first 100 tosses? What is your net gain or loss at this point?

Fill in the Blanks

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

a. What percentage of times has heads come up in the first 100 tosses? What is your net gain or loss at this point?

Multiple Choice

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

b. Suppose you toss the coin 200 more times (a total of 300 tosses), and at that point heads has come up 47% of the time. Is this increase in the percentage of heads consistent with the law of large numbers? Explain. What is your net gain or loss at this point?

• Follows law of large numbers.

• Does not follow.

Fill in the Blanks

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

b. Suppose you toss the coin 200 more times (a total of 300 tosses), and at that point heads has come up 47% of the time. Is this increase in the percentage of heads consistent with the law of large numbers? Explain. What is your net gain or loss at this point?

b. Suppose you toss the coin 200 more times (a total of 300 tosses), and at that point heads has come up 47% of the time. Is this increase in the percentage of heads consistent with the law of large numbers? Explain. What is your net

0.47 (300) = 141 won

0.53 (300) = 159 lost

subtract for loss of 18 dollars

Fill in the Blanks

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

c. How many heads would you need in the next 100 tosses in order to break even after 400 tosses? Is this likely to occur?

(Currently we have 141 heads.)

Is is likely to get 59 heads out of 100 flips?

Possible, but not likely.

Open Ended

23. Gambler’s Fallacy and Coins. 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, heads comes up 46 times and tails comes up 54 times.

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.

Open Ended

27. Lottery Draw. Consider a lottery game in which six balls are drawn randomly from a set of balls numbered 1 through 42. One week, the winning combination consists of balls numbered 5, 12, 23, 32, 36, and 41. The next week, the winning balls are numbered 1, 2, 3, 4, 5, and 6. Is the second winning set more or less likely than or just as likely as the first? Explain.

Fill in the Blanks

31. House Edge in Blackjack. In a large casino, the house wins on its blackjack tables with a probability of 50.7%. All bets at blackjack are 1 to 1: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

a. If you bet $1 on each hand, what is the expected value to you of a single game? What is the house edge?

(tenth of a cent)(three decimals)

31. House Edge in Blackjack. In a large casino, the house wins on its blackjack tables with a probability of 50.7%. All bets at blackjack are 1 to 1: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

a. If you bet $1 on each hand, what is the expected value to you of a single game? What is the house edge?

• Player : $1 (0.493) - $1(0.507)

• = -$0.014

• Casino: -$1(0.493) + $1(0.507)

• = $0.014

Fill in the Blanks

31. House Edge in Blackjack. In a large casino, the house wins on its blackjack tables with a probability of 50.7%. All bets at blackjack are 1 to 1: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

b. If you played 100 games of blackjack in an evening, betting $1 on each hand, how much should you expect to win or lose? Explain.

(Expected value of -$0.014 per $1 game.)

Fill in the Blanks

31. House Edge in Blackjack. In a large casino, the house wins on its blackjack tables with a probability of 50.7%. All bets at blackjack are 1 to 1: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

c. If you played 100 games of blackjack in an evening, betting $5 on each hand, how much should you expect to win or lose? Explain.

(Expected value of -$0.014 per $1 game.)

Fill in the Blanks

31. House Edge in Blackjack. In a large casino, the house wins on its blackjack tables with a probability of 50.7%. All bets at blackjack are 1 to 1: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

d. If patrons bet $1,000,000 on blackjack in one evening, how much should the casino expect to earn? Explain.

(Expected value of -$0.014 per $1 game.)

Fill in the Blanks

33–34: Powerball. The table below gives prizes and probabilities of winning (on a single $1 ticket) for the Multi-State Powerball lottery.

33. Find the expected value of the winnings for a single lottery ticket if the jackpot is $30 million. How much can you expect to win or lose each year if you buy 10 lottery tickets per week? Should you actually expect to win or lose this amount? Explain.

(nearest cent)

Fill in the Blanks

33–34: Powerball. The table below gives prizes and probabilities of winning (on a single $1 ticket) for the Multi-State Powerball lottery.

33. Find the expected value of the winnings for a single lottery ticket if the jackpot is $30 million. How much can you expect to win or lose each year if you buy 10 lottery tickets per week? Should you actually expect to win or lose this amount? Explain.

(Expected value : -$0.58)