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Existing User? Log in to At The Races. New User? Register Now It's free. 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. As the events are independent, the result is. If A and B are mutually exclusive, then if event A happens, B cannot.
Also, if event B happens, then A cannot. If A and B are independent, then the outcome of A has no effect on the outcome of B, and the outcome of B has no effect on the outcome of A. Their respective outcomes have no effect on each other. Q: Do both events have to be independent? Can one event be independent and the other dependent? Q: Are all games on a roulette wheel independent? A: Yes, they are. Separate spins of the roulette wheel do not influence each other.
In each game, the probabilities of the ball landing on a red, black, or green remain the same. How do I use a Venn diagram to tell if events are independent? Venn diagrams are great if you need to examine intersections and show mutually exclusive events. The Case of the Two Classes. The Head First Health Club prides itself on its ability to find a class for everyone.
As a result, it is extremely popular with both young and old. The Health Club is wondering how best to market its new yoga class, and the Head of Marketing wonders if someone who goes swimming is more likely to go to a yoga class. The CEO disagrees.
They ask a group of 96 people whether they go to the swimming or yoga classes. Out of these 96 people, 32 go to yoga and 72 go swimming. Are the yoga and swimming classes dependent or independent? Well, I hear you keep getting fledgling statisticians into trouble. You want to work out the probability of getting two independent events? Just multiply the probabilities for the two events together and job done.
With independent events, probabilities just turn out the same. I think that people need to think of me first instead of you; that would sort out all of these problems. By really thinking through whether events are dependent or not. Let me give you an example. Suppose you have a deck of 52 cards, and thirteen of them are diamonds. What would be the probability of that happening? The events are dependent.
You can no longer say there are 13 diamonds in a pack of 52 cards. Not fair, I assumed you put the first card back! That would have meant the probability of getting a diamond would have been the same as before, and I would have been right. The events would have been independent.
When people think about you first, it leads them towards making all sorts of inappropriate assumptions. Solved: The Case of the Two Classes. So how do we know the classes are independent? Here are a bunch of situations and events.
Your task is to say which of these are dependent, and which are independent. Choosing a card from a deck of cards, putting the card back in the deck, and then choosing another one. Your task was to say which of these are dependent, and which are independent. When you remove one sock, there are fewer socks to choose from the next time, and this affects the probability. Besides the chances of winning, you also need to know how much you stand to win in order to decide if the bet is worth the risk.
Betting on an event that has a very low probability may be worth it if the payoff is high enough to compensate you for the risk. Fred decides to throw a coin. I person eats alone if Fred and George go to the Diner. As these events are independent, this is equal to P 22 x P Skip to main content. Head First Statistics by Dawn Griffiths. Start your free trial. Calculating Probabilities: Taking Chances.
Roll up for roulette! Your very own roulette board. Note Just be careful with those scissors. Place your bets now! Brain Power What things do you need to think about before placing any roulette bets? What are the chances?
Vital Statistics: Event An outcome or occurrence that has a probability assigned to it. Sharpen your pencil. Sharpen your pencil Solution. Find roulette probabilities. You can visualize probabilities with a Venn diagram. Complementary events. BE the croupier Solution. Note The most likely event out of all these is that the ball will land in a black pocket. There are no Dumb Questions. Q: Q: Why do I need to know about probability? Q: Q: Are probabilities written as fractions, decimals, or percentages?
A: A: They can be written as any of these. A: A: There certainly is. Q: Q: Do I always have to draw a Venn diagram? A: A: No. And the winning number is You can also add probabilities. Vital Statistics: Probability To find the probability of an event A, use.
Q: Q: It looks like there are three ways of dealing with this sort of probability. A: A: It all depends on your particular situation and what information you are given. Q: Q: If some events are so unlikely to happen, why do people bet on them? A: A: A lot depends on the sort of return that is being offered. Q: Q: Does adding probabilities together like that always work? A: A: Think of this as a special case where it does. You win! Time for another bet. Note Uh oh!
Different answers. Exclusive events and intersecting events. Brain Power What sort of effect do you think this intersection could have had on the probability? Problems at the intersection. Some more notation. Watch it! BE the probability Solution. Exercise Solution. A: A: Yes it is. Q: Q: Is there a limit on how many events can intersect?
Another unlucky spin Conditions apply. Find conditional probabilities. Note The probability of A give that we know B has happened. You can visualize conditional probabilities with a probability tree. Trees also help you calculate conditional probabilities.
Handy hints for working with trees. Probability Magnets Solution. You can get coffee with or without donuts. Note Hint: maybe some of your other answers can help you.
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