# What Is the Probability Rule

If you`ve been able to answer these questions correctly, you probably have a good instinct to calculate probability! Read on to find out how we will apply this knowledge. Therefore, the probability that he will choose the green tie, gray shirt and black pants is 15×16×14=1120dfrac{1}{5} times dfrac{1}{6} times dfrac{1}{4} = boxed{dfrac{1}{120}}51×61×41=1201. The probability that she (a) will consult a work of fiction is 0.40, (b) a non-fiction book is 0.30, and (c) fiction and non-fiction are 0.20. How likely is it that the student will consult a fiction book, a non-fiction book, or both? The fundamental error results from a misunderstanding of the conditional probability and neglect of the previous probability that a defendant would be guilty before that evidence was presented. If a prosecutor has gathered evidence (p. ex. B a DNA match) and an expert testifies that the probability of concluding that evidence if the accused were innocent is minimal, the error occurs when it is concluded that the probability that the accused is innocent must be relatively low. If DNA matching is used to confirm guilt that is otherwise suspected, then it is indeed solid evidence. However, if DNA evidence is the only evidence against the defendant and the defendant was selected from a large database of DNA profiles, the likelihood that the match will be random may be reduced.

Therefore, it is less prejudicial to the defendant. Probabilities in this scenario are not related to the probability of being guilty; They refer to the chances of being selected at random. Note that if [latex]text{A}[/latex] and [latex]text{B}[/latex] are independent, we have that [latex]text{P}(text{B}|text{A})= text{P}(text{B})[/latex], so the formula becomes [latex]text{P}(text{A} cap text{B})=text{P}(text{A})text{P}(text{B})text{P}(text{B})[/latex], which we encountered in a previous section. As an example, consider the experience of rolling a dice and throwing a coin. The probability that we will get a [latex]2[/latex] on the cube and a tail on the part is [latex]frac{1}{6}cdot frac{1}{2} = frac{1}{12}[/latex], since the two events are independent of each other. Motivation question for Rule 2: A person in the United States is randomly selected. How likely is it that the person will have blood type A? Before discussing the rules of probability, let`s give the following definitions: Here is a summary of the rules we have presented so far. Let`s review what we`ve learned so far. We can calculate any probability in this scenario if we can determine how many people encounter the event or combination of events. It is important to note that each matrix role is independent.

That is, if the result of the first punching roll is 6, it does not affect the probability that the second punching roller will lead to 6. They have three cubes with 6 sides. One is red, the other is blue and the other is green. What is the probability that you will roll a 1 on the red cube, a 2 on the blue cube and a 3 on the green cube? Since event A and event “non A” together constitute all possible outcomes, and since rule 2 tells us that the sum of probabilities of all possible outcomes is 1, the following rule should be quite intuitive: Solution: Leave F = the event that the student verifies the fiction; and leave N = the event where the student consults non-fiction books. Then, depending on the addition rule: Remember that rule 4, the addition rule, has two versions. One is limited to the disjoint events we`ve already covered, and we`ll discuss the more general version later in this module. The same goes for probabilities when TWO friends play billiards and decide to flip a coin to determine who will play first in each round. In the first two towers, the coin lands on heads. They decide to play a third round and turn the coin over again. What is the probability that the coin will land on the heads again? This is the (funny) news in the clip of the Daily Show that we provided on the previous page. But let`s think again. In this clip, Walter claims that since there are two possible outcomes, the probability is 0.5.

The two possible outcomes are then P(A or B or C) = P(A) + P(B) + P(C). The rule applies to any number of unrelated events. We`ve also given you some tools to help you find the probabilities of events – namely the rules of probability. Suppose you select two cards “with a replacement” by returning your first card to the deck and shuffling the deck before selecting the second card. Since the deck of cards is complete for both selections, the first selection does not affect the probability of the second selection. When selecting cards with replacements, the selections are independent. In a previous lesson, we learned two important properties of probability: Using the product rule for dependent events, the probability that the error decreases after 30 seconds is 1×23×13= 291times dfrac{2}{3}timesdfrac{1}{3}=boxed{dfrac{2}{9}}1×32×31=92 According to the Blood Center (bloodcenter.stanford.edu) at Stanford University, These are the probabilities of human blood groups in the United States (the probability for type A was intentionally omitted): The product rule is a guideline of when probabilities can be multiplied to create another significant probability. Specifically, the product rule is used to determine the probability of overlapping events: you have probably noticed that the probability section is significantly different from the previous two sections. It has a much larger technical/mathematical component, so the results are more of the “good or bad” nature.

If two events are mutually exclusive, then the probability that one of the two will occur is the sum of the probabilities of each event. See if you can answer the following questions using the charts and/or list of results for each event as well as what you`ve learned about probabilities so far. A VERY common error is the incorrect application of the multiplication rule for independent events that are discussed on the next page. This is only true if A and B are independent (see definitions below), which is rarely the case with data represented in bidirectional tables. If the ghost opens one of these 4 rooms, how likely is it that he will not see anyone hiding there? Consider a good roll of the dice, which is another example of independent events. If one person rolls two, the result of the first role does not change the probability of the outcome of the second role. Suppose we select three children at random and are interested in the likelihood that none of the children will have birth defects. Fortunately, these rules are very intuitive, and as long as they are applied systematically, they will allow us to solve more complex problems; especially problems for which our intuition might be inadequate. How likely is it that a randomly selected person will not be able to donate blood to everyone? In other words, what is the probability that a randomly selected person will not have blood type O? We have to find P (not O).

Using the complement rule, P(not O) = 1 – P(O) = 1 – 0.44 = 0.56. In other words, 56% of the US population does not have blood type O: There are 5 possible routes (red, blue, green, yellow, purple) from Candy Castle to the chocolate forest and 3 routes (pink, white, orange) from the chocolate forest to Ice Cream Cottage, if each path is chosen at random, how likely is it that you will travel from Candy Castle via the purple and orange road to Ice Cream Cottage? According to the product rule× there is a chance of 15×13=115dfrac{1}{5} times dfrac{1}{3} =boxed{dfrac{1}{15}}51×31=151 to choose the purple-orange path. If you can calculate a probability using logic and counting, you do not NEED a probability rule (although the correct rule can always be applied) The probability that an event will inform us of the probability of its occurrence can vary from 0 (indicating that the event will never happen) to 1 (indicating that the event is safe). Bayes` rule is commonly used in statistics, science and engineering, para. B example in: model selection, probabilistic expert systems based on Bayesian networks, statistical evidence in court proceedings, spam filters, etc. Bayes` rule tells us how unconditional and conditional probabilities are related, whether we are working with a frequentist or Bayesian interpretation of probability. According to the Bayesian interpretation, it is often used in the situation where [latex]text{A}_1[/latex] and [latex]text{A}_2[/latex] are competing assumptions and [latex]text{B}[/latex] are observed evidence. The rule shows how one`s own judgment as to whether [latex]text{A}_1[/latex] or [latex]text{A}_2[/latex] is true needs to be updated when observing the evidence. b. If A and B are independent – neither event affects or affects the probability that the other event will occur – then P(A and B) = P(A)*P(B). This particular rule extends to more than two independent events. Example: P(A and B and C) = P(A)*P(B)*P(C) ExampleAn urn contains 6 red marbles and 4 black marbles.

Two marbles are extracted from the urn without replacement. What is the probability that both marbles are black? There is a probability of 15dfrac{1}{5}51 that Calvin will randomly select the green tie, a probability of 16dfrac{1}{6}61 that he will randomly select the gray shirt, and a probability of 14dfrac{1}{4}41 that he will randomly select the black pants. Later, we will discuss the rules for calculating P (A and B). The bug has 5 adjacent edges to choose from within the first 10 seconds, but it doesn`t matter which one it chooses as they all bring it closer to the ground.. .