Probability in Genetics: Multiplication and Addition Rules

Introduction to Multiplication and Addition in Genetics

• Presenter: Mr. Andersen

• Topic: Application of multiplication and addition in genetics

• Note: Though these concepts seem simple, their application to genetics can be complex.

Multiplication Rule

• Definition: The multiplication rule occurs when dealing with independent events that are going to occur in sequence.

• Key Indicator: The use of the word "and" in a question suggests the multiplication rule should be applied.

Coin Flip Example

• Example Question: What are the odds of flipping a coin five times and getting tails on every flip?

• Probability of tails in one flip:

  • P(tails) = $\frac{1}{2}$

• Calculation:

  • Multiply probabilities of each flip:

  • $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32}$

• Conclusion:

  • Probability of getting tails every time = $\frac{1}{32}$

Example of Children’s Gender

• Question: What are the odds of all five children being boys?

• Probability of each child being a boy:

  • P(boy) = $\frac{1}{2}$

• Total Probability Calculation:

  • Multiply probabilities:

  • $\left(\frac{1}{2}\right)^5 = \frac{1}{32}$

Roll Dice Example

• Question: What are the odds of rolling snake eyes (two 1s) on two dice?

• Probability for one die:

  • P(1) = $\frac{1}{6}$

• Calculation:

  • $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

• Insight: Elicits an event that occurs less than 3% of the time.

Card Example

• Question: What are the odds of picking a three and then a king from a deck of cards?

• Probability of picking a three (returning to the deck):

  • P(three) = $\frac{4}{52} = \frac{1}{13}$

• Probability of picking a king (after replacing the three):

  • P(king) = $\frac{4}{52} = \frac{1}{13}$

• Total Probability:

  • $\frac{1}{13} \times \frac{1}{13} = \frac{1}{169}$

Application to Genetics: Mendel's Work

Example Problem

• Scenario: Cross between pea plants that are heterozygous for purple flowers.

• Asking for probability of homozygous recessive offspring:

• Parental Genotypes:

  • Parent 1 = Pp (big P little p)

  • Parent 2 = Pp (big P little p)

• Probability for Analyzing Homozygous Recessive (pp):

  • P(contributing p from Parent 1) = $\frac{1}{2}$

  • P(contributing p from Parent 2) = $\frac{1}{2}$

• Calculation:

  • Total Probability = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Trihybrid Cross

• Scenario: Probing deeper, asking about specific offspring from a trihybrid cross.

• Common Student Reaction: Fear of constructing large Punnett squares.

• Steps to Simplify:

  1. Analyze one trait at a time (letter by letter).

  2. Example Question: Probability of getting specific combinations.

  3. Probability of As:

  • Parent A: $\frac{1}{2}$ to get offspring (Big A or little a).

  1. Repeat for Bs and Cs:

  • P(B) and P(C) similarly calculated = $\frac{1}{2}$.

  1. Combine results:

  • Final Calculation = $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$

Addition Rule

• Definition: The addition rule applies to mutually exclusive events.

• Key Indicator: The presence of the word "or" in a question indicates the addition rule should be applied.

Coin Flip Example

• Question: What are the odds of getting either heads or tails?

• Individual Probabilities:

  • P(heads) = $\frac{1}{2}$

  • P(tails) = $\frac{1}{2}$

• Total Probability Calculation:

  • Combine:

  • $\frac{1}{2} + \frac{1}{2} = 1$ (or 100%).

Rolling a Die Example

• Question: What are the odds of rolling a 2 or a 5?

• Individual Probabilities:

  • P(2) = $\frac{1}{6}$

  • P(5) = $\frac{1}{6}$

• Total Probability Calculation:

  • Combine:

  • $\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

Card Example

• Question: What's the odds of picking either a 5 or a heart?

• Individual probabilities:

  • P(5) = $\frac{4}{52}$

  • P(heart) = $\frac{13}{52}$

• Adjust for Overlap:

  • Eliminate the 5 of hearts from the count:

  • Effective hearts = 12.

• Total Probability Calculation:

  • Combine Probabilities:

  • $\frac{4 + 12}{52} = \frac{16}{52} = \frac{4}{13}$

Genetic Example

• Scenario: What are the odds of getting heterozygous offspring from a cross between two heterozygous pea plants?

• Calculation Methods:

  1. Possible combinations of big P and little p.

  2. Probability when combined by two distinct paths:

     1. Big P from one parent, little p from another.

     2. Little p from one parent, big P from another.

  • Combine results using addition rule:

  • $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

Final Example: Integration of Rules

• Cross between two parents, both heterozygous:

• Looking for odds of obtaining specific traits. Each combination analyzed piecewise:

  • Combine utilizing multiplication where independent events exist and addition when seeking either/or events.

• Result: Analytical understanding of genetic probabilities increases accuracy while simplifying complex problems.

Summary

• Multiplication is used for events occurring together (independent events).

• Addition is for mutually exclusive events (either/or situations).

• Using these rules can considerably ease calculations in genetic problems, avoiding cumbersome Punnett squares.

• Encouragement to practice problems for better understanding of application in real-world genetics scenarios.