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Tree Diagram Probability Tree Diagram Probability Tree diagrams can be a helpful way of organizing outcomes in order to identify probabilities. For example, if we have a box with two red, two green and two white balls in it, and we choose two balls without looking, what is the probability of getting two balls of the same color? P(same color) = P(RR or GG or WW) We use the tree diagram to the left to help us identify the possible combinations of outcomes. Here we see that there are nine possible outcomes, listed to the right of the tree diagram. This number is the size of the sample space for this two state experiment, and will be in the denominator of each of our probabilities. Each of these possible nine outcomes has a probability of 1/9, which we can find using the multiplication rule P(RR or GG or WW) = 3/9. EXAMPLES: TWO FLIPS OF A FAIR COIN Consider the two-step experiment ‘two flips of a fair coin’. Let H denote Head, and T denote Tail. Know More About Types of Bias

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Note that the sample space for this experiment is: S={HH,HT,TH,TT} The probability tree diagram is shown at right. From the tree diagram, we can easily compute various probabilities: P(both heads)=P(HH)=(0.5)(0.5)=0.25 P(both tails)=P(TT)=(0.5)(0.5)=0.25 P(no heads)=P(TT)=0.25 P(at least one head) Example: A bag contains 3 black balls and 5 white balls. Paul picks a ball at random from the bag and replaces it back in the bag. He mixes the balls in the bag and then picks another ball at random from the bag. a) Construct a probability tree of the problem. b) Calculate the probability that Paul picks: i) two black balls ii) a black ball in his second draw

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Define Mutually Exclusive Define Mutually Exclusive Definition of Mutually Exclusive Events Two or more events are said to be mutually exclusive if they cannot occur at the same time. In other words, events that have no outcomes in common are said to be mutually exclusive events or disjoint events. If two events A and B are mutually exclusive, then the probability of the occurrence of A or B is the sum of the their individual probabilities. A mutually exclusive event is one in which the acceptance of one alternative automatically excludes other, possible alternatives. A common example of a mutually exclusive event is a coin flip. Either the coin will come up heads or tails. Since the coin coming up heads means that it will not come up tails, that makes the coin flip a mutually exclusive event. It is either will be one side or the other, it cannot be both. Law :- The field of law is highly conscious of mutually exclusive events. While this is true of many crimes, a commonplace scenario would be receiving a speeding ticket. Either the person was exceeding the speed limit, or they were not. This is a simple example, but often the entire basis of guilt or innocence is based on a mutually exclusive event, hence the importance of an alibi. If it can be proven a defendant was doing something else at the time of the crime then they cannot be guilty of the crime (in many cases).

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Math :- The mathematical definition of mutually exclusive events is a little more involved. According to mathematics, mutually exclusive events are when two or more events cannot occur at the same time, and the sum of their individual probabilities is the chance of the event occurring at all. This adds one more element to the definition, as either one or another event must happen, but both events cannot happen at the same time. Formula :- The mathematical formula for determining the probability of mutually exclusive events is P(A U B) = P(A) + P(B). Spoken out loud the formula is "if A and B are mutually exclusive events, then the probability of A or B occurring is equal to the probability of event A plus the probability of event B." Probability :- Mutually exclusive events are one of the keystones of probability. Probability is the chance that a certain event will happen a certain amount of the time. For instance, flipping a coin is a mutually exclusive event that has a 50 percent probability--about half the time it will be heads, about half the time it will be tails. In statistics and regression analysis, an independent variable that can take on only two possible values is called a dummy variable. For example, it may take on the value 0 if an observation is of a male subject or 1 if the observation is of a female subject. The two possible categories associated with the two possible values are mutually exclusive, so that no observation falls into more than one category, and the categories are exhaustive, so that every observation falls into some category. Sometimes there are three or more possible categories, which are pairwise mutually exclusive and are collectively exhaustive — for example, under 18 years of age, 18 to 64 years of age, and age 65 or above. In this case a set of dummy variables is constructed, each dummy variable having two mutually exclusive and jointly exhaustive categories — in this example, one dummy variable (called D1) would equal 1 if age is less than 18, and would equal 0 otherwise; a second dummy variable (called D2) would equal 1 if age is in the range 18-64, and 0 otherwise. Read More About Examples of Bias

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Tree Diagram Probability