Probability

Exam Information

  • Exam #3 Details:

    • Date: In-class on Friday, November 7th

    • Allowed Items: Portable computers or devices, lecture slides, homework questions, textbook, crossword, YouTube videos

    • Students with Accommodations: Must contact DRES to schedule an exam time

    • Instructor: Add Dr. Braz as the instructor

    • Office Hours Announcements provided

Power and Probability in Genetics

Source Material

  • Textbook: "U Campbell Biology, 12th Edition"

  • Chapter: 14 (Concepts 14.2)

Core Concepts

  • Learning Objectives:

    • Describe the rules of probability

    • Calculate simple probabilities

    • Explain how the rules of probability can be used to solve complex genetic problems

    • Apply probability to real-world scenarios

Probability Basics

  • Definition: Probability calculations are used in genetic problems to predict the outcome of crosses.

    • The probability of an event is the likelihood that the event will occur in the future.

    • Formula:

    • \text{Probability of an event} = \frac{\text{Number of times an event occurs}}{\text{Total number of events}}

    • This also represents the ratio of specific events to the total number of events.

Examples of Probability

Coin Tossing

  • Example:

    • The chance of tossing heads is \frac{1}{2}

    • The probability of tossing tails is \frac{1}{2}

Card Draw

  • Example:

    • A standard deck has 52 cards.

    • The probability of picking the ace of spades is \frac{1}{52}

    • The chance of picking a card other than the ace of spades is \frac{51}{52}

Fundamental Probability Properties

  • Range of Probability:

    • Probability ranges from 0 to 1

    • \text{Prob (event) = 1} indicates an event that always occurs

    • \text{Prob (event) = 0} indicates an event that never occurs

    • All possible outcomes for an event must add up to 1.

    • Example: \text{Prob (Heads)} + \text{Prob (Tails)} = \frac{1}{2} + \frac{1}{2} = 1

Independent Events

  • Definition: Segregation in a heterozygous individual resembles flipping a coin concerning the calculation of probability for each outcome.

  • Example: Tossing two coins:

    • Each toss is independent of the other toss.

    • Outcomes of any toss do not affect previous trials.

Rules of Probability

Multiplication Rule

  • Definition: The probability that two or more independent events will occur together is equal to the product of their respective probabilities.

    • For two events:

    • \text{Prob (event A and event B)} = \text{Prob (event A)} \times \text{Prob (event B)}

  • Genetic Example:

    • For a plant to have wrinkled seeds (rr):

    • \text{Prob (rr)} = \text{Prob (r)} \times \text{Prob (r)}

    • \text{Prob (rr)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Addition Rule

  • Definition: The probability that two or more mutually exclusive events will occur is the sum of their individual probabilities.

    • For two events:

    • \text{Prob (event A or event B)} = \text{Prob (event A)} + \text{Prob (event B)}

  • Genetic Example:

    • For an offspring to be heterozygous from heterozygous parents (Bb x Bb):

    • \text{Prob (Rr)} + \text{Prob (rR)} = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}

Probabilities in Genetic Crosses

Example: Cattle Coat Color

  • Scenario: Coat color in cattle where B (black coat, dominant) and b (red coat, recessive).

    • Question: Probability of two heterozygous parents (Bb x Bb) having a red offspring:

    • \text{Prob (bb)} = \text{Prob (b)} \times \text{Prob (b)}

    • \text{Prob (bb)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Tracking Inheritance

Example: Dihybrid Cross for Cattle

  • Scenario: Two heterozygous parents (Bb x Bb) where:

    • B allele = black coat; b allele = red coat

    • Question: Probability of black offspring

    • \text{Prob (BB or Bb)} = \text{Prob (BB)} + \text{Prob (Bb)}

    • \text{Prob (BB or Bb)} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4}

Multiple Traits: Dihybrid Cross

  • Scenario: Coat color in dogs with alleles B (black, dominant) and b (red, recessive), and D (pigment, dominant) and d (dilute, recessive).

    • Question: Probability dihybrid parents (BbDd x BbDd) having a black offspring:

  • Apply multiplication and addition rules to form the solution.

Genetic Cross Scenarios

Complex Scenarios

  • Example:

    • Tracking inheritance of three characters in a cross: PpYyRr x Ppyyrr

    • Probabilities for offspring with at least two recessive traits are calculated based on individual probabilities for each event.

    • Probabilities for offspring genotypes such as \text{Prob (ppyyRr)}, etc., are deduced from parent gametes.

General Insights

  • The rules of probability provide the predicted likelihood of various genetic outcomes.

  • The larger the sample size, the closer results conform to predictions.

Practice

Home Exercise

  • Example Calculate probabilities when tossing two coins:

  • \text{Prob (HH)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

  • \text{Prob (TT)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

  • \text{Prob (HT)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

  • \text{Prob (TH)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

  • Explore results of tossing coins 8 times for further analysis.

Summary of Learning Content

  • Recap of learning objectives, including the description and application of probability rules within genetic contexts.