Lab 6

Objectives

1. Practice with Probability

Rules of Probability

Tip1. Addition Rule:

\[ P(A \cup B) = P(A) + P(B) - P(A\cap B) \]

Tip2. Compliment Rule:

\[ P(A) + P(A^C) = 1 \] A little algebra gives us,

\[ P(A^C) = 1 - P(A) \]

Tip3. Multiplication Rule:

\[ \begin{align} P(A \cap B) &= P(A)P(B|A) \\ \\ &= P(B)P(A|B) \end{align} \]

We can solve for conditional probabilities:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Tip4. Law of Total Probability

\[ P(A) = P(A \cap B) + P(A \cap B^C) \]

Tip5. Bayes Rule

\[ P(A|B) = \frac{P(A) P(B|A)}{P(A)P(B|A) + P(A^C)P(B|A^C)} \]

Additional Notation:

1. Intersections, unions, compliments slide 8

Practice

1. Football

On the Iowa Football Team there are exactly 99 players on the team. Of them 32 are freshman, 12 are sophomores, 34 are juniors, and 21 are seniors.

1. Are the years of study mutually exclusive?

2. If a football player is selected at random what is the probability that he is not a senior?

3. If you select a random football player and it is given that he is not a senior what is the probability of him being a sophomore?

4. What is the probability that I pull two juniors out randomly with replacement? Without replacement?

2. Periodontal Status

Periodontal status refers to gum disease where individuals are classilfied as follows: healthy, gingivitis, periodontal disease (perio).

1. Given the following probabilities, construct a table of counts that has the form below.

	Periodontal Status
Gender	Healthy	Gingivitis	Perio	Total
Male
Female
Total

Probabilties:

• \(P(\text{Person is male}) = 0.3749\)

• \(P(\text{Person is male and healthy}) = 0.1429\)

• \(P(\text{Person is male and has Perio}) = 0.1167\)

• \(P(\text{Person is healthy}) = 0.4672\)

• \(P(\text{Person is female and has Perio}) = 0.1147\)

Hint: choose 1000 for your total.

Complete Table1

	Periodontal Status
Gender	Healthy	Gingivitis	Perio	Total
Male	143	115	117	375
Female	324	186	115	625
Total	467	301	232	1,000

2. Calculate the following conditional probabilities.

   • P(Healthy | Female)

   • P(Male | Perio)

   • P(Perio or Gingivitis | Female)

3. Chameleon Color Disease

A rare genetic condition called Chromatic Drift Syndrome (CDS) affects 2% of chameleons in a large population. A diagnostic test for CDS has:

• Sensitivity = 85%, (85% of chameleons with CDS test positive)

• False positive rate = 12%, (12% of healthy chameleons test positive)

If a chameleon tests positive, what is the probability that it actually has CDS?

4. Pangolin Migration Patterns

Researchers are studying pangolins in a large wildlife reserve.

Each year:

• 40% of pangolins migrate to a new feeding area during the dry season.

• 60% stay in their original territory.

Researchers also observe feeding success:

• If a pangolin migrates, the probability it gains enough weight for the winter is 0.75.

• If a pangolin does not migrate, the probability it gains enough weight is 0.45.

  1. What is the overall probability that a randomly selected pangolin gains enough winter weight?

  2. Given that a pangolin gained enough weight, what is the probability it migrated?

  3. Are migration and gaining enough weight independent events?

Handwritten solutions