Expected value is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it’s one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.

• Gambling And Odds Quizlet 5

Gambling - Gambling - Chances, probabilities, and odds: Events or outcomes that are equally probable have an equal chance of occurring in each instance. In games of pure chance, each instance is a completely independent one; that is, each play has the same probability as each of the others of producing a given outcome.

1. Online & Vegas sports betting odds & lines, betting news & picks for 2020. Covers the most trusted source of sports betting information since 1995.
2. The best hand (because of the low probability that it will occur) is the royal flush, which consists of 10, J, Q, K, A of the same suit.There are only 4 ways of getting such a hand (because there are 4 suits), so the probability of being dealt a royal flush is.

Example 1

In the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?

Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you’ve lost $2. On a per-space basis, you have “won” –$2/$38 ≈ –$0.053. In other words, on average you lose 5.3 cents per space you bet on.

We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).

There is another way to compute expected value without imagining what would happen if we play every possible space. There are 38 possible outcomes when the wheel spins, so the probability of winning is
[latex]displaystylefrac{{1}}{{38}}[/latex].

Summarizing these along with the values, we get this table:

Notice that if we multiply each outcome by its corresponding probability we get $35 ×

[latex]displaystylefrac{{1}}{{38}}[/latex] = –0.9737, and if we add these numbers we get 0.9211 + (–0.9737) ≈ –0.053, which is the expected value we computed above.

Expected Value

Expected Value is the average gain or loss of an event if the procedure is repeated many times.

We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.

Try it Now 1

You purchase a raffle ticket to help out a charity. The raffle ticket costs $5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth $4000. Compute the expected value for this raffle.

Example 2

In a certain state’s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, then win $1,000. It costs $1 to buy a ticket. Find the expected value.

Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:

[latex]displaystylefrac{{{}_{{6}}{C}_{{6}}}}{{{}_{{48}}{C}_{{6}}}}=frac{{1}}{{12271512}}approx{0.0000000815} text{ for all 6 numbers,}[/latex]

[latex]displaystylefrac{{{left({}_{{6}}{C}_{{5}}right)}{left({}_{{42}}{C}_{{1}}right)}}}{{{}_{{48}}{C}_{{6}}}}=frac{{252}}{{12271512}}approx{0.0000205} text{ for 5 numbers}[/latex]

Our probabilities and outcome values are:

The expected value, then is:
[latex]displaystyle{left(${999},{999}right)}timesfrac{{1}}{{12271512}}+{left(${999}right)}timesfrac{{252}}{{12271512}}+{left(-${1}right)}timesfrac{{12271259}}{{12271512}}approx-${0.898}[/latex]

On average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1.

In general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money. It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the
average winnings are positive it could be the case that most people lose money and one very fortunate individual wins a great deal of money. If the expected value of a game is 0, we call it a fair game, since neither side has an advantage.

Not surprisingly, the expected value for casino games is negative for the player, which is positive for the casino. It must be positive or they would go out of business. Players just need to keep in mind that when they play a game repeatedly, their expected value is negative. That is fine so long as you enjoy playing the game and think it is worth the cost. But it would be wrong to expect to come out ahead.

Try it Now 2

A friend offers to play a game, in which you roll 3 standard 6-sided dice. If all the dice roll different values, you give him $1. If any two dice match values, you get $2. What is the expected value of this game? Would you play?

Expected value also has applications outside of gambling. Expected value is very common in making insurance decisions.

Example 3

According to the estimator at
numericalexample.com, a 40-year-old man in the US has a 0.242% risk of dying during the next year. An insurance company charges $275 for a life-insurance policy that pays a $100,000 death benefit. What is the expected value for the person buying the insurance?

The probabilities and outcomes are

The expected value is ($99,725)(0.00242) + (–$275)(0.99758) = –$33.

Not surprisingly, the expected value is negative; the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.

For people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.

1. [latex]displaystyle{left(${3},{995}right)}timesfrac{{1}}{{2000}}+{left(-${5}right)}timesfrac{{1999}}{{2000}}approx-${3.00}[/latex]
2. Suppose you roll the first die. The probability the second will be different is [latex]displaystylefrac{{5}}{{6}}[/latex].

   The expected value is
   [latex]displaystyle{left(${2}right)}timesfrac{{16}}{{36}}+{left(-${1}right)}timesfrac{{20}}{{36}}=frac{{12}}{{36}}approx${0.33}[/latex]. Yes, it is in your advantage to play. On average, you’d win $0.33 per play.

David Lippman, Math in Society, “Probability,” licensed under a CC BY-SA 3.0 license.

Heather R, ”
Roulette,” licensed under a CC BY-SA 3.0 license.

Many people will be visiting a casino this summer, whether on a trip to Vegas, on a cruise ship, or just at a local casino during a 'staycation.'

But before you go, you may want to know which games give you the best chance of winning.

So we went behind the scene at two casinos, and spoke with the game managers, to find out which games give you an edge.

Table Games Your Best Bet

Jay Bean, a Caesar's floor manager, agreed to sit down with us and talk about the games with the best odds.

He said many people don't realize that table games have much better odds than slots. They are intimidated by thoughts of card sharks taking their money, with images of an intense Texas Hold 'Em tournament in their minds, and so they stick with the safe games: the slots.

But they are doing themselves a disservice, he said.

Bean's advice? After you've spent some time on the slot machines, move on to the Blackjack table, and start by just observing.

1. Blackjack

Blackjack has the best odds of winning, with a house edge of just 1 percent in most casinos, Bean said.

Plus, you are playing against only the dealer, not hooded poker champions.

'Blackjack is one of our easiest games to play,' Bean said. 'You're just looking for a number that beats the dealer's number without going over 21.'

If you're a novice, Bean said to sit down during a less busy time, such as the afternoon, and the dealer will walk you through the game.

But the same is not true with poker: You should practice the game with your friends first, at home, Bean said, until you are comfortable with it. Bean said novices are best off passing on the poker tables, where yes, the competition can be tough.

You can find out more information about the game of blackjack at the following link:
http://wizardofodds.com/games/blackjack/

2. Craps

So from the Blackjack table, he suggests moving on to the dice game Craps, the game with the second best odds, also nearly 50-50.

The Craps table can be a bit intimidating for the beginner with all the boxes on the table, Bean admitted. But it's really not tough, and it has one of your best chances of winning. In the end, all you are really doing is betting on a dice roll.

For more information on craps, visit:
http://wizardofodds.com/games/craps/

3. Roulette

Bean said your next stop should be the Roulette wheel. It's simple, and pays fairly well.

'So you bet on your favorite numbers,' Bean said. 'The dealer spins, and if your number comes in you win.'

If you stick to betting on just Reds or Blacks, you have nearly a 50-50 chance of walking away a winner. If it lands on red when you choose red, you will double your bet, Bean explained.

While it may seem you have an even chance of winning by simply choosing a color, there is one catch that gives the house its edge: the extra green 0 and 00 position on the wheel. It's on all American roulette wheels.

Or, if you feel lucky, play just one number. It's hard to win, but if you do it pays out 36 to 1.

For more information on roulette, visit:
http://wizardofodds.com/games/roulette/

Worst Odds

Your worst odds? The casinos, understandably, did not want to talk about games with poor odds.

So we turned to Forbes Magazine, which in a report about Best Bets at the Casino, said two popular games -- the Wheel of Fortune and the ever-popular slot machines -- have the lowest odds of winning, with a house edge of 10 percent or more.

The Experts Agree

Professor of Analytics Michael Magazine has crunched the numbers, and agreed that blackjack is the best way to go home with money in your pockets. He said if you take a little time to learn the game, you have the best odds of winning.

'You are somewhere between 44 percent to 48 percent in every single hand of winning,' he said, noting that no other casino game gives you odds that good.

But Magazine cautioned that he's not promoting gambling, because he said even games with the best odds still favor the house.

'Play it long enough and you will lose. In the long term the house doesn't lose,' Magazine said.

If You Love The Slots....

Finally, if you really enjoy playing the slots, here is some good information to know, from state casino reports we examined.

With slots, the more money you put in per bet, the higher your chance of winning. If you put the max in a penny machine, which is usually around 50 cents, you have much better odds than if you put in 2 cents.

In addition, penny slots have the lowest odds, with $1 and $5 slots paying much better.

Always gamble responsibly, and don't waste your money.
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You can learn more about specific odds of slots machines in each state that has casino gambling by visiting the American Casino Guide.

Don't Waste Your Money is a registered trademark of the EW Scripps Co.

Gambling And Odds Quizlet 5

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