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# Binomial Distributions

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Practice Binomial Distributions
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# Binomial Distributions - Answer Key

## Probability and STDs

### Topic

Probability and STDs

### Vocabulary

• Binomial Distribution
• Binomial Probabilities
• Binompdf Function

### Student Exploration

#### If you are at a party hanging out with friends, what are the chances that the people you are talking with have an STD?

Many times adults warn kids not to have sex because of diseases that you can get from sex. An infection that can be acquired through sexual contact (in blood, semen, vaginal or other bodily fluids) is called a Sexually Transmitted Disease (STD). See this website for more information on specific STDs, http://www.mayoclinic.com/health/sexually-transmitted-diseases-stds/DS01123.Are warnings from adults fear tactics to scare kids to not have sex? Or are they true?

If you are at a party hanging out with friends, how likely is it that the people in your group have an STD? You can use binomial distributions to measure the likelihood of the people in your group having an STD. But first you need some information on different STDs.

Use the following website from the Center for Disease Control (CDC) to research the percentage of young people with different STDs. Note that the slide for the young adults only includes statistics on percentages of young adults with Chlamydia and Gonorrhea, and by gender (men and women). Check the other links for information on different STDs.

Choose at least two different STDs and find the percentage of young people that have each STD (in your area, if that information is available) from the sites above. Note you might only be able to find separate statistics for boys and girls, that is okay but just indicate that in your answers.

The answers below are for Chlamydia in woman between the ages of 15 and 19. In the United States 3,378.2 women out of 100,000 women have Chlamydia. Thus the probability of a woman having Chlamydia is0.033782 or 3.38%.

1. Then use your knowledge of binomial distributions to calculate the probability that if you were at a party talking with a group of ten people, what is the probability that exactly one person in the group would have the first STD? What is the probability that exactly one person in the group would have the second STD? The probability of exactly one woman out of ten having Chlamydia is calculated by: $P(X = 1) = {_{10}}C_1 (0.033782)^1 (0.966218)^{10} =10(0.033782)(0.709171442) = 0.23957 \approx 24.0%$
2. Then use your knowledge of binomial distributions to calculate the probability that if you were at a party talking with a group of ten people, what is the probability that exactly two people in the group would have the first STD? What is the probability thatexactlytwo people in the group would have the second STD? The probability of exactly two women out of ten having Chlamydia is calculated by: $P(X = 2) = {_{10}}C_2 (0.033782)^2 (0.966218)^8 =45(0.0011412235)(0.7596280459) = 0.03901 \approx 3.9%$
3. Then use your knowledge of binomial distributions to calculate the probability that if you were at a party talking with a group of ten people, what is the probability that one or two people in the group would have the first STD? What is the probability that one or two people in the group would have the second STD? The probability that one or two women in the group have Chlamydia is the sum of your answers in #1 and 2,$P(X = 1) = 0.23957 \approx 24.0%$ and $P(X = 2) = 0.03901 \approx 3.9%$.$P(X = 1 \ or \ 2) = 0.23957 + 0.03901 \approx 24.0% + 3.9% = 27.9%$.
4. Then use your knowledge of binomial distributions to calculate the probability that if there were 75 people at the party, what is the probability that exactly two people at the party would have the first STD? What is the probability that exactly five people at the party would have the first STD? The probability of exactly two women out of 75 having Chlamydia is calculated by: $P(X = 2) = {_{75}}C_2 (0.033782)^2 (0.966218)^{73} = 2775(0.0011412235)(0.0813736957) = 0.2577019674\approx 25.8%$
5. Calculate the probability that if there were 75 people at the party, what is the probability that exactly five people at the party would have the second STD? What is the probability that exactly five people at the party would have the second STD? The probability of exactly five women out of 75 having Chlamydia is calculated by: $P(X = 5) = {_{75}}C_5 (0.033782)^5 (0.966218)^{70} = 17,259,390(0.000000043997)(0.090210828) = 0.0685032063\approx 6.9%$
6. Now analyze your answers. How does the probability change based on the number of people in the group you are calculating for? Explain this in detail. How do the probabilities of the different STDs compare? Explain. The smaller the number of people from a population, the more likely it is that they have Chlamydia. As the number of people gets bigger, it becomes less that they have Chlamydia.
7. Going back to the original questions, are warnings from adults about sex fear tactics? Or are they true? Has this activity changed the way you think about the likelihood that a person might have an STD? Why or why not.

Do you not live in the United States? Choose a city in the United States that you are somewhat familiar with, such as Los Angeles, New York, San Francisco,Chicago, Washington D.C. etc. Then complete the same activity. If you’re country has a government organization that provides similar information as the CDC in the US, use that resource

### Extension Investigation

Use the website above to investigate how the probability of people having one STD change based on their gender, race, sexual orientation, or geographic location. For example if you want to analyze the STD by gender, calculate the probability of men and women (separately) in a group would have the STD. For example if you want to analyze the STD by race, calculate the probability of people of one race and people of a different race (separately) in a group that would have the STD.

Now analyze your answers. How does the probability change based on their gender, race, sexual orientation, or geographic location? Explain this in detail.

### Connections to other CK-12 Subject Areas

• Binomial Distributions
• Binomial Distributions and Probability
• Combinations
• Combinations Problems
• Theoretical and Experimental Probability
• Probability and Permutations