Applying the Multiplication Rule for Probability

Transcription

1 Knowledge Article: Probability and Statistics Applying the Multiplication Rule for Probability Independent Events and the Multiplication Rule If you flip a coin once, you will get one of two outcomes: heads or. However, if you toss two coins, there are four possible outcomes: heads-heads, heads-, -heads, and -. In the first case, there is a single event: tossing one coin. On the other hand, the second example involves two events: tossing two coins. The second event is a combination of two events. Such an event is called a compound event. Each event in a compound event has its own individual probability. How do we find the combined probability of the compound event using the individual probabilities? If the events are independent, the answer is simple. We multiply the individual probabilities of the constituent events to get the probability of the compound event. This rule is known as the multiplication rule. Note that the multiplication rule can be applied only in cases of independent events. The multiplication rule may be written mathematically as P(A and B) = P(A) P(B), where A and B are independent events. An independent event is an event whose occurrence has no effect on the probability of the occurrence of any other event. Stated mathematically, events A and B are independent if any one of the following is true: P(A B) = P(A) P(B A) = P(B) P(A and B) = P(A) P(B) Let s consider an example. You know that if you flip a coin (a single independent event) the chance of getting heads is 50%. In the language of probability, we say that the probability is ½. If you flip two coins (two independent events), there are four possible outcomes: heads-heads, heads-, -heads, and -. It's reasonable to ask whether heads- is the same outcome as -heads: the answer is that it isn t. Even if you don t keep track of the order in which you flip the coins, heads- is still a different result from -heads, so each must be counted separately. So, in this case, the chance of getting heads-heads is.

2 What if you flip a coin three times (three independent events)? In this case, there are eight possible results. The table shows a systematic way of listing all eight results. The chance of getting three heads in a row is 8. First Coin Second Coin Third Coin End Result heads heads heads HHH HHT heads HTH HTT heads heads THH THT heads TTH TTT By now, a pattern might be apparent to you: adding a coin to the sequence doubles the number of possibilities. Let s look at a slightly more complicated, and more interesting, example, the SongWriter. You are the proud inventor of the SongWriter 000 and have trademarked it. A user sets the song speed (fast, medium, or slow); volume (loud or quiet), and style (rock or country). Then, the SongWriter automatically writes a song to match. How many possible settings are there? You might guess that the answer is + + = 7, but, in fact, there are more. You can see them all in this tree diagram. If you start at the top of a diagram tree and move down, you end up with one particular kind of song, for instance, fast, loud, country song. There are different song types in all. This comes from multiplying the number of settings for each knob: =.

3 Now, suppose the machine has a randomize setting that randomly chooses the speed, volume, and style. What is the probability that you will end up with a loud rock song that is not slow? To answer a question like this, you can use the following process. Count the total number of results (the leaves in the tree) that match your criteria. In this case there are two: the fast-loud-rock path and the medium-loudrock path. Count the total number of results. As you just saw, there are. Divide. The probability of a loud rock song that is not slow is, or 6. Note that this process will always give you a number from 0 (no results match) through (all results match), inclusive. Probabilities always have values from 0 (for something that never happens) through (for something that is guaranteed to happen), inclusive. But what does it really mean to say that the probability is 6? You aren t going to get 6 of a song. One way to make this result more concrete is to imagine that you run the machine on the randomize setting 00 times. You should expect to get loud rock songs that are not slow one out of every six times; roughly 7 songs will match that description. This gives us another way to express the answer: there is a 7% probability of any given song matching this description. We can look at this problem another way. What is the chance that any given randomly selected song will not be slow? The answer is. This means that, ideally, two out of every three randomly chosen songs will not be slow. Now, out of those, how many will be loud? Half of them. The probability that a randomly selected song is both not slow and loud is, or. And now, out of that, how many will be rock? Again, half of them:. This leads us back to the conclusion we came to earlier: 6 of all randomly chosen songs will be loud rock songs that are not slow. But it also gives us an example of a very

4 general principle that is at the heart of all probability calculations: When two events are independent and do not occur simultaneously, the probability that they will both occur is the probability of the first multiplied by the probability of the other. What does it mean to describe two events as independent? It means that they have no effect on each other. In real life, we know that rock songs are more likely to be fast and loud than slow and quiet. Our machine, however, keeps all three categories independent: choosing rock does not make a song more likely to be fast or slow, loud or quiet. In some cases, applying the multiplication rule is very straightforward. Suppose you generate two different songs: what is the chance that they will both be fast songs? The two songs are independent of each other, so the chance is. 9 Suppose you generate five different songs. What s the chance that they ll all be fast? It is, or chance in. Not very likely, as you might suspect! Other cases are less obvious. Suppose you generate five different songs. What is the probability that none of them will be a fast song? The multiplication rule tells us only how to find the probability of this and that; how can we apply it to this question? The key is to reword the question, like this: what is the chance that the first song will not be fast, and the second song will not be fast, and the third song will not be fast, and so on? Expressed in this way, the question is a perfect candidate for the multiplication rule. The probability of the first song not being fast is. Same for the second song, and so on. So the probability is, or roughly %. Based on this, we can easily answer another question: if you generate five different songs, what is the probability that at least one of them will be fast? Once again, the multiplication rule does not apply directly here; it tells us this and that, not this or that. But we can recognize that this is the opposite of the previous question. We said that % of the time, none of the songs will be fast. That means that the other 87% of the time, at least one of them will. Dependent Events (Conditional Probability) An event that occurs if and only if another event has occurred before it is called a dependent event. The occurrence of the last event is affected by the occurrence of the preceding events and is dependent on it.

5 Recall that the multiplication rule is applicable only to independent events. How do we find the combined probability of dependent events? We find the combined probability of dependent events by using the concept of conditional probability. Assume that event B happens before event A. The conditional probability of A given B is written P(A B). P(A B) is the probability that event A will occur given that the event B has already occurred. A conditional probability reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A B) is P( A and B) P( A B), where P(B) > 0. P( B) Let's work through an example. What is the probability that two cards drawn at random, one after another, without replacement, from a deck of 5 playing cards will both be aces? If you just use the formula for the combined probability of two independent events, the answer you get is However, this is incorrect, because the two events here are not independent. If the first card drawn is known to be an ace, then the probability that the second card is also an ace must be less than because there are only three aces left in the deck after the 5 first card has been drawn. The probability that the second card is an ace given that the first card is an ace is an example of a conditional probability. In this case, the condition is that the first card is an ace. Symbolically, we can write this as: P(ace on second draw ace on first draw). The symbol is read as "given," so the above expression is a short way to say, "the probability that an ace is drawn on the second draw given that an ace was drawn on the first draw." What is this probability? If an ace is drawn on the first draw, there are three aces left in a deck of 5 cards. This means that the probability that one of these aces will be drawn is If events A and B are not independent, then P(A and B) = P(A) P(B A). Applying this general formula to the problem here, the probability of drawing two aces from a deck as stated is

6 If events A and B are independent, P(B A) = P(B) as the occurrence of B does not depend on the occurrence of A. The formula then reduces to the original multiplication rule, P(A and B) = P(A) P(B). This knowledge article is adapted from the following sources:. Felder, Kenney M. Advance Algebra II: Conceptual Explanations. Connexions. March 9, 00. Lane, David M., Project Leader, Rice University. Online Statistics Education: A Multimedia Course of Study. 6