1 / 17

190 likes | 727 Views

Binomial Formula, Mean, and Standard Deviation. Presentation 5.3. A Binomial Situation. Suppose a couple plans to have 3 children. The chance they have a boy is 0.2. The gender of one child is independent of the gender of another child. Let X be the number of boys they have.

Download Presentation
## Binomial Formula, Mean, and Standard Deviation

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Binomial Formula, Mean, and Standard Deviation**Presentation 5.3**A Binomial Situation**• Suppose a couple plans to have 3 children. • The chance they have a boy is 0.2. • The gender of one child is independent of the gender of another child. • Let X be the number of boys they have.**Binomial Specifics**• If the chance a child is a boy is 0.2, what’s the chance a child is a girl? • The chance the child is a girl is 0.8. • How many gender sequences (i.e. BBB, BBG, BGG, etc) are possible? • There are 8. • BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG**We want to fill in the probability distribution below:**P(X=0) and P(X=3) are already given from the results on the previous page. We only need P(X=1) and P(X=2) now.**Binomial Formula**• P(X = 1) = P(only one boy) = P(GGB) + P(GBG) + P(BGG) = [(0.8)2(0.2)] + [(0.8)2(0.2)] + [(0.8)2(0.2)] = 3 (0.8)2(0.2)**Binomial Distribution**Similarly, P(X=2) equals 3(0.2)2(0.8). Now we can complete the probability distribution of X.**Binomial Formula**• P(X = 1) = P(one boy and two girls) = P(GGB) + P(GBG) + P(BGG) = [(0.8)2(0.2)] + [(0.8)2(0.2)] + [(0.8)2(0.2)] = 3 (0.8)2(0.2) The above form can be generalized into the binomial formula.**Binomial Formula**• The formula is: • This is for n trials • The probability of success is p • The probability of failure is 1-p • We are interested in k successes • Then there are n-k failures**Binomial Formula**The probability of all the successes happening. The probability of all the failures happening. Number of ways to get success. This is a combination of n choose k. On the TI calculator, this is under Math, Prb and nCr.**Binomial Formula**• What is the probability of having all boys when having 3 children? • What is the probability of having 2 boys and 1 girl when having 3 children?**Binomial Formula**• What is the probability of having 1 boy and 2 girls when having 3 children? • What is the probability of having all girls when having 3 children?**Example**The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that precisely 7 jurors are black. Clearly, n=12 and p=.6, so You can confirm this with the calculator by trying: Binompdf (12, .6, 7)**The mean, variance and standard deviation**• μ= np • σ 2= np(1-p) • If we have a sample of 20 light bulbs and 5% of all bulbs produced are defective, then on average np=1 bulb will be defective.**Example**The AP Statistics Exam contains 40 multiple choice questions each with 5 possible answers. What kind of score might you expect if you randomly guess on each question. Let x be a random variable defined by x = number of correct answers on such an exam Find the mean and standard deviation for x**Example - solution**The random variable is clearly binomial with n = 40 and p = 0.2 The mean and standard deviation of x are You should expect to get 8 questions correct**Binomial Formula, Mean, and Standard Deviation**• This concludes this presentation.

More Related