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Section 5. The focus of the section was on discrete probability distributions pdf. To find the pdf for a situation, you usually needed to actually conduct the experiment and collect data. Then you can calculate the experimental probabilities. Normally you cannot calculate the theoretical probabilities instead. However, there are certain types of experiment that allow you to calculate the theoretical probability. One of those types is called a Binomial Experiment. If you know you have a binomial experiment, then you can calculate binomial probabilities.
This is important because binomial probabilities come up often in real life. Examples of binomial experiments are:. Suppose you are given a 3 question multiple-choice test. Each question has 4 responses and only one is correct. Suppose you want to find the probability that you can just guess at the answers and get 2 questions right.
Teachers do this all the time when they make up a multiple-choice test to see if students can still pass without studying. To help with the idea that you are going to guess, suppose the test is in Martian. To answer this question, start with the sample space. The same is similar for the other outcomes. What you did in chapter four was just to find three divided by eight.
However, this would not be right in this case. That is because the probability of getting a question right is different from getting a question wrong. What else can you do? Look at just P RRW for the moment. To find the probability of 2 correct answers, just add these three probabilities together. You get. You could go through the same argument that you did above and come up with the following:.
Hopefully you see the pattern that . You can now write the general formula for the probabilities for a Binomial experiment. Be careful, a success is not always a good thing. Sometimes a success is something that is bad, like finding a defect. A success just means you observed the outcome you wanted to see happen. Binomial Formula for the probability of r successes in n trials is. When solving problems, make sure you define your random variable and state what n, p, q, and r are. Without doing this, the problems are a great deal harder. Consider a group of 20 people.
At most three means that three is the highest value you will have. Find the probability of x is less than or equal to three. The reason the answer is written as being greater than 0. It is best to write the answer as greater than 0. At least four means four or more. Find the probability of x being greater than or equal to four. That would mean adding up all the probabilities from four to twenty. This would take a long time, so it is better to use the idea of complement.
The complement of being greater than or equal to four is being less than four. That would mean being less than or equal to three. Part e has the answer for the probability of being less than or equal to three. Just subtract that from 1. Actually the answer is less than 0. Since the probability of finding four or more people with green eyes is much less than 0. If this is true, then you may want to ask why Europeans have a higher proportion of green-eyed people. That of course could lead to more questions.
The binomial formula is cumbersome to use, so you can find the probabilities by using technology. If you have the new software on the TI, the screen looks a bit different. Thus there is an Again, you will use the binompdf command or the dbinom command.
Following the procedure above, you will have binompdf 20,. The probability that out of twenty people, nine of them have green eyes is a very small chance. Your answer is 0. Thus there is a really good chance that in a group of 20 people at most three will have green eyes. Again use binomcdf or pbinom. Again there is a really good chance that at most two people in the room will have green eyes. Suppose you consider a group of 10 children. Notice, the answer is given as 0.
The event of five or more is improbable, but not impossible. See solutions, b. See solutions, c. See solutions. Properties of a binomial experiment or Bernoulli trial Fixed of trials, n , which means that the experiment is repeated a specific of times. The n trials are independent, which means that what happens on one trial does not influence the outcomes of other trials.
There are only two outcomes, which are called a success and a failure. Examples of binomial experiments are: Toss a fair coin ten times, and find the probability of getting two he. Question twenty people in class, and look for the probability of more than half being women? Shoot five arrows at a target, and find the probability of hitting it five times? What is the random variable? Is this a binomial experiment? What is the probability of getting 2 questions right? What is the probability of getting zero right, one right, and all three right?
Solution a. There are 3 questions, and each question is a trial, so there are a fixed of trials. Getting the first question right has no affect on getting the second or third question right, thus the trials are independent. Either you get the question right or you get it wrong, so there are only two outcomes. In this case, the success is getting the question right.
The probability of getting a question right is one out of four. This is the same for every trial since each question has 4 responses. State the random variable. Argue that this is a binomial experiment. Find the probability that none have green eyes. Find the probability that nine have green eyes. Find the probability that at most three have green eyes. Find the probability that at most two have green eyes. Find the probability that at least four have green eyes. In Europe, four people out of twenty have green eyes. Is this unusual?
What does that tell you? There are 20 people, and each person is a trial, so there are a fixed of trials. Either a person has green eyes or they do not have green eyes, so there are only two outcomes. In this case, the success is a person has green eyes. The probability of a person having green eyes is 0. This is the same for every trial since each person has the same chance of having green eyes. Find the probability that none have autism. Find the probability that seven have autism. Find the probability that at least five have autism.
Find the probability that at most two have autism. Suppose five children out of ten have autism. There are 10 children, and each child is a trial, so there are a fixed of trials. If you assume that each child in the group is chosen at random, then whether has autism does not affect the chance that the next child has autism.
Thus the trials are independent. Either has autism or they do not have autism, so there are two outcomes. In this case, the success is has autism. This is the same for every trial since each child has the same chance of having autism. Consider a grouping of fifteen people. Argue that this is a binomial experiment Find the probability that None are left-handed. Seven are left-handed.
At least two are left-handed. At most three are left-handed. At least seven are left-handed.Looking for the one out of 5
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