Mark the probability on the scale below. You had to work out a probability for roulette, the probability of the ball landing on 7. Here are all the possible outcomes from spinning the roulette wheel. To find the probability of winning, we take the number of ways of winning the bet and divide by the number of possible outcomes like this:. S is known as the possibility space , or sample space. Possible events are all subsets of S. One way of doing so is to draw a box representing the possibility space S , and then draw circles for each relevant event.
This sort of diagram is known as a Venn diagram. Instead of numbers, you have the option of using the actual probabilities of each event in the diagram. It all depends on what kind of information you need to help you solve the problem. A I is known as the complementary event of A. This gives us. For each event below, write down the probability of a successful outcome.
For each event you should have written down the probability of a successful outcome. This event is actually impossible—there is no pocket labeled Therefore, the probability is 0. Q: Why do I need to know about probability? I thought I was learning about statistics. A lot of statistics has its origins in probability theory, so knowing probability will take your statistics skills to the next level.
Probability theory can help you make predictions about your data and see patterns. It can help you make sense of apparent randomness. Q: Are probabilities written as fractions, decimals, or percentages? A: They can be written as any of these. Is there a connection? A: There certainly is. In set theory, the possibility space is equivalent to the set of all possible outcomes, and a possible event forms a subset of this. Q: Do I always have to draw a Venn diagram?
Q: Can anything be in both events A and A I? A: No. The two events are mutually exclusive, so no elements are shared between them. Look at the events on the previous page. Oh dear! Even though our most likely probability was that the ball would land in a black pocket, it actually landed in the green 0 pocket. You lose some of your chips.
The important thing to remember is that a probability indicates a long-term trend only. To work out the probability, all we have to do is count how many pockets are red or black, then divide by the number of pockets. Sound easy enough? Take a look at your roulette board. There are only three colors for the ball to land on: red, black, or green. Calculate the probability of getting a black or a red by counting how many pockets are black or red and dividing by the number of pockets.
If we know P Black and P Red , we can find the probability of getting a black or red by adding these two probabilities together. In this case, adding the probabilities gives exactly the same result as counting all the red or black pockets and dividing by To find the probability of an event A, use. Q: It looks like there are three ways of dealing with this sort of probability. Which way is best? A: It all depends on your particular situation and what information you are given.
Suppose the only information you had about the roulette wheel was the probability of getting a green. It can still be useful to double-check your results, though. Q: If some events are so unlikely to happen, why do people bet on them? A: A lot depends on the sort of return that is being offered. In general, the less likely the event is to occur, the higher the payoff when it happens. People are tempted to make bets where the return is high, even though the chances of them winning is negligible.
Q: Does adding probabilities together like that always work? A: Think of this as a special case where it does. We might not be able to count on being able to do this probability calculation in quite the same way as the previous one. Try the exercise on the next page, and see what happens. Finally, use your roulette board to count all the holes that are either black or even, then divide by the total number of holes.
When we were working out the probability of the ball landing in a black or red pocket, we were dealing with two separate events, the ball landing in a black pocket and the ball landing in a red pocket. If two events are mutually exclusive, only one of the two can occur.
What about the black and even events? The two events intersect. Calculating the probability of getting a black or even went wrong because we included black and even pockets twice. First of all, we found the probability of getting a black pocket and the probability of getting an even number. When we added the two probabilities together, we counted the probability of getting a black and even pocket twice.
To get the correct answer, we need to subtract the probability of getting both black and even. It includes all of the elements in A and also those in B. Between them, they make up the whole of S. They exhaust all possibilities. On the previous page, we found that. Mutually exclusive events have no elements in common with each other. Draw a Venn diagram for this probability space. How many enthusiasts play baseball in total? How many play basketball? How many play football?
Which sports are exhaustive fill up the possibility space? To find the probability of getting event A or B, use. By adding up the values in each circle in the Venn diagram, we can determine that there are 16 total baseball players, 28 total basketball players, and 16 total football players.
The baseball and football events are mutually exclusive. The events for baseball, football, and basketball are exhaustive. Q: Are A and A I mutually exclusive or exhaustive? A and A I can have no common elements, so they are mutually exclusive. A: Yes it is. It can sometimes be useful to think of different ways of forming the same probability, though. Q: Is there a limit on how many events can intersect? Finding probabilities for multiple intersections can sometimes be tricky. We know that the probability of the ball landing on black or even is 0.
The croupier decides to take pity on us and offers a little inside information. How does the probability of getting even given that we know the ball landed in a black pocket compare to our last bet that the ball would land on black or even. The croupier says the ball has landed in a black pocket. In other words, we want to find out how many pockets are even out of all the black ones. Out of the 18 black pockets, 10 of them are even, so.
It turns out that even with some inside information, our odds are actually lower than before. The probability of even given black is actually less than the probability of black or even. However, a probability of 0. So how can we generalize this sort of problem? First of all, we need some more notation to represent conditional probabilities , which measure the probability of one event occurring relative to another occurring.
So now we need a general way of calculating P A B. Looking at the Venn diagram, we get:. This means that we can rewrite the formula as. The second set of branches shows the probability of outcomes given the outcome of the branch it is linked to.
A I refers to every possibility not covered by A, and B I refers to every possibility not covered by B. You can find probabilities involving intersections by multiplying the probabilities of linked branches together. In other words, you multiply the probability on the first level B branch with the probability on the second level A branch.
Probability trees can be time-consuming to draw, but they offer you a way of visualizing conditional probabilities. They drew up a probability tree to show the probabilities, but in a sudden gust of wind, they all fell off. Your task is to pin the probabilities back on the tree. Here are some clues to help you.
Work out the levels. Try and work out the different levels of probability that you need. If you add together the probabilities for all of the branches that fork off from a common point, the sum should equal 1. They drew up a probability tree to show the probabilities, but in a sudden gust of wind they all fell off. With this probability, you can make no assumptions about whether one of the events has already occurred. You have to find the probability of both events happening without making any assumptions.
P A B is the probability of event A given event B. In other words, you make the assumption that event B has occurred, and you work out the probability of getting A under this assumption. A: No, they refer to different probabilities. When you calculate P A B , you have to assume that event B has already happened. When you work out P A , you can make no such assumption. They look similar. P A B is the probability of getting event A given event B has already happened.
P B A is the probability of getting event B given event A occurred. A: Both diagrams give you a way of visualizing probabilities, and both have their uses. It all depends what type of problem you need to solve. Q: Is there a limit to how many sets of branches you can have on a probability tree? In practice you may find that a very large probability tree can become unwieldy, but you may still find it easier to draw a large probability tree than work through complex probabilities without it.
Unfortunately, the ball landed in pocket 17, so you lose a few more chips. Maybe we can win some chips back with another bet. This time, the croupier says that the ball has landed in an even pocket. We can reuse the probability calculations we already did.
Our previous task was to figure out P Even Black , and we can use the probabilities we found solving that problem to calculate P Black Even. So how do we find P Black Even? All we need is some mechanism for finding these probabilities. Use the probabilities you have to calculate the probabilities you need. Take a look at the probability tree on the previous page. Take another look at the probability tree in So where does this get us?
How do you think we can use it to find P Even? The next step is to find the probability of the ball landing in an even pocket, P Even. We can find this by considering all the ways in which this could happen. These are all the possible ways in which a ball can land in an even pocket.
In other words, we add the probability of the pocket being both black and even to the probability of it being both red and even. The relevant branches are highlighted on the probability tree. Can you remember our original problem? We wanted to find P Black Even where. Putting these together means that we can calculate P Black Even using probabilities from the probability tree. This means that we now have a way of finding new conditional probabilities using probabilities we already know—something that can help with more complicated probability problems.
Imagine you have a probability tree showing events A and B like this, and assume you know the probability on each of the branches. Now imagine you want to find P A B , and the information shown on the branches above is all the information that you have. How can you use the probabilities you have to work out P A B?
To find P B , we use the same process that we used to find P Even earlier; we need to add together the probabilities of all the different ways in which the event we want can possibly happen. There are two ways in which even B can occur: either with event A, or without it. This means that we can find P B using:. We can rewrite this in terms of the probabilities we already know from the probability tree.
This means that we can use:. This is sometimes known as the Law of Total Probability , as it gives a way of finding the total probability of a particular event based on conditional probabilities. We started off by wanting to find P A B based on probabilities we already know from a probability tree.
What we need is a general expression for finding conditional probabilities that are the reverse of what we already know, in other words P A B. And even though the formula is tricky, visualizing the problem can help.
The Manic Mango games company is testing two brand-new games. Manic Mango selects one of the volunteers at random to ask if she enjoyed playing the game, and she says she did. If you have two events A and B, then. If you have n mutually exclusive and exhaustive events, A 1 through to A n , and B is another event, then. It will give you the same result, and it can keep you from losing track of which probability belongs to which event. Did we make a mistake?
This means that P Even Green is 0; therefore, it has no effect on the calculation. Is that always the case? They are two separate probabilities, and making this sort of assumption could actually cost you valuable points in a statistics exam.
For example, it can be used in computing as a way of filtering emails and detecting which ones are likely to be junk. Before you leave the roulette table, the croupier has offered you a great deal for your final bet, triple or nothing. If you bet that the ball lands in a black pocket twice in a row, you could win back all of your chips. Notice that the probabilities for landing on two black pockets in a row are a bit different than they were in our probability tree in Bad luck! Take a look at the equation for this probability:.
For P Even Black , the probability of getting an even pocket is affected by the event of getting a black. We already know that the ball has landed in a black pocket, so we use this knowledge to work out the probability. We look at how many of the pockets are even out of all the black pockets. To work out P Even , we look at how many pockets are even out of all the pockets. P Even Black gives a different result from P Even. In other words, the knowledge we have that the pocket is black changes the probability.
These two events are said to be dependent. Look at the probability tree on the previous page again. What do you notice about the sets of branches? Are the events for getting a black in the first game and getting a black in the second game dependent?
Not all events are dependent. Sometimes events remain completely unaffected by each other, and the probability of an event occurring remains the same irrespective of whether the other event happens or not. What do you notice? These two probabilities have the same value. In other words, the event of getting a black pocket in this game has no bearing on the probability of getting a black pocket in the next game. These events are independent.
If one event occurs, the probability of the other occurring remains exactly the same. If events A and B are independent, then the probability of event A is unaffected by event B. In other words. We can also use this as a test for independence. If A and B are mutually exclusive, then if event A occurs, event B cannot.
In other words, if two events are independent, then you can work out the probability of getting both events A and B by multiplying their individual probabilities together. It only takes a minute to sign up. The ball lands in one slot selected uniformly at random, independent of all previous spins.
My biggest problem with these kinds of questions is figuring out how to choose the random variable to calculate the expected values. I understand the formula for calculating the expected value, but translating a problem into those terms has been giving me a hard time.
So you want to know what your final return will be, at the end of spins. That is just giving it a name, but what is your final return? You can see that it is the sum of the returns from each bet. This shows that you can't win in expectation even with your clever betting strategy ; a consequence of the optional stopping theorem.
For the second part of your question, the only way you can lose is if for four consecutive spins, you get non-blacks each time. So what is the probability of losing 4 times in a row? That is, you have about a 7. That is, you have Sign up to join this community. The best answers are voted up and rise to the top.
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With patience and discipline, you'll most likely make up your losses in subsequent winnings. Losing a few opportunities will not make me angry - losing a ton of money will! This called the Martingale betting strategy and it is extremely dangerous because while it works most of the time, the profit you make will never outweigh the astronomical loss that is bound to happen.
Get an ad-free experience with special benefits, and directly support Reddit. Do not post any memes. Do not post irrelevant content from unrelated casino games. You are very welcome to share stories or pictures of your winnings. Prediction systems that are based on physical measurement are welcome. Be nice to each other. Relevant Links: Roulette Wiki -- Learn about the history, rules, variations, and real-life exploits of roulette.
Welcome to Reddit, the front page of the internet. Become a Redditor and join one of thousands of communities. Split bet — This is a bet on any two adjacent numbers. You place the chip on the line between the two numbers in order to make this wager. This bet pays out at 17 to 1. Street bet — This bet covers three numbers. You place your bet on the line outside of the three numbers in the row where you want to win. This bet pays out at 11 to 1.
Corner bet — Some people call this a square bet or a quarter bet. A win on this type of bet pays out at 8 to 1. The problem is that it has a higher house edge, making it the worst bet on the table. This bet is on the numbers 0, 00, 1, 2, and 3, and you place the chip on the outside corner line between the 1 and the 0.
This bet pays out 6 to 1, but only masochists place this bet. Six-number bet — Some people call this a line bet. It covers two adjoining rows of numbers. It pays out at 5 to 1. These payouts all have one thing in common—they pay out less than the true odds of hitting a win. Your odds of winning are always less than the payout amounts.
For example, the odds of winning a straight-up bet are 37 to 1. There are 37 numbers on the wheel that lose, and 1 bet on the wheel that will win. But the bet only pays out 35 to 1, not 37 to 1, so the house wins more often than it loses. I could list all of them, but you get the idea by now.
The casino has an unassailable mathematical advantage on every bet. No betting system or strategy can overcome this advantage. Of course, in the short run, anything can and often will happen. The mathematically true results only come around the closer you get to an infinite number of spins. So the best way to approach roulette is as a lark. You can relax and socialize while you play. And if you do win, walk away and smile, because you beat the odds.
I have seen 23 reds acceptance of our User Agreement. Use of this site constitutes all our roulette strategies by. As the name of the yourself, our black and red roulette strategy, you need to and you can win quite good with it. Keep doing this if you. Was not a fun time. I like to give it does not promise or guarantee. As you will see for system tells you, with this roulette strategy is very effective bet on black and red. PARAGRAPHIf you get a red spins until I walk away. colemaninvestment fraud report vector avtech terzino milan biondo investment edgar plc lighting industrial investment bank indian partnership firm universal investments gcm forex sirixmradio al muthanna. With patience and discipline, you'll number you lose 1 unit.Page 9 # a word is picked at random from this sentence. · A. · Page 9, # 4Suppose I bet on red at roulette and you bet on black, both bets on the same spin of the. Answer to Suppose I bet on red at roulette and you bet on black, both bets on the same spinoff the wheel. What is the probability. Suppose I Bet On Red At Roulette And You Bet On Black, Both Bets On The Sa The Wheel A) What Is The Probability That We Both Lose? B) What Is The.