Calculate probability ratios in offspring — Hinglish notes

3.1.9 · Biology › Mendelian Genetics

WHY do we even need probability?

Poora topic probability ke do laws par tika hua hai jo seedhe Mendel se aate hain:

HOW the rules come from gamete-making

Monohybrid $3 : 1$ ratio khud derive karo. Cross $A a \times A a$ :

	$A$ (½)	$a$ (½)
$A$ (½)	$AA$ = ¼	$A a$ = ¼
$a$ (½)	$A a$ = ¼	$aa$ = ¼

$P (AA) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ (product rule)
$P (A a) = A from mom \frac{1}{2} \cdot \frac{1}{2} + A from dad \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{2}$ (sum rule on two ways)
$P (aa) = \frac{1}{4}$

Dominant phenotype $= P (AA) + P (A a) = \frac{1}{4} + \frac{1}{2} = \frac{3}{4}$ . Recessive $= \frac{1}{4}$ . Phenotype ratio $= 3 : 1$ , genotype ratio $= 1 : 2 : 1$ — derived, yaad nahi kiya.

Figure — Calculate probability ratios in offspring

HOW to handle many genes at once — the fork-line method

Do heterozygotes ke cross mein har gene ke liye: $P (dominant phenotype) = \frac{3}{4}, P (recessive phenotype) = \frac{1}{4}$ $P (homozygous dom AA) = \frac{1}{4}, P (hetero A a) = \frac{1}{2}, P (homo rec aa) = \frac{1}{4}$

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Socho har parent ke paas ek bag hai jisme har trait ke liye do colored marbles hain. Woh bina dekhe andar haath daalte hain aur baby ko ek marble dete hain — bilkul random, 50/50. Baby ke ek specific pair of marbles milne ka chance jaanne ke liye, aap poochho "mom se kya chance?" times "dad se kya chance?" — kyunki dono hone chahiye, aap multiply karte ho. Agar aap do mein se kisi bhi result se khush ho, toh aap unhe add karte ho. Bas itni hi genetics probability hai: marbles ko bags se nikalne ka ek fancy game.

Flashcards

What does the product rule state and when is it used?

P (A and B) = P (A) \times P (B)

; independent events ke liye use hota hai jo dono hone chahiye (e.g., mom's aur dad's gametes combine karna).

What does the sum rule state and when is it used?

P (A or B) = P (A) + P (B)

; mutually exclusive alternative outcomes ke liye use hota hai (e.g.,

A a

do tarike se banna).

Why is

P (A a) = \frac{1}{2}

but

P (AA) = \frac{1}{4}

in an

A a \times A a

cross?

A a

do tarike se ban sakta hai (A-from-mom/a-from-dad OR a-from-mom/A-from-dad), isliye

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

;

AA

sirf ek hi tarike se banta hai.

A a \times A a

, what is the genotype ratio and phenotype ratio?

Genotype

1 : 2 : 1

(

AA : A a : aa

); phenotype

3 : 1

(dominant:recessive).

A a B b \times A a B b

, what is

P

of showing both dominant traits?

\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}

How do you compute "at least one" probability?

Complement use karo:

P (at least one) = 1 - P (none)

For

A a B b C c \times A a B b C c

, what is

P (AA bb C c)

\frac{1}{4} \times \frac{1}{4} \times \frac{1}{2} = \frac{1}{32}

When does the product rule across genes FAIL?

Jab genes linked hoon (same chromosome par) aur independently assort na karein.

Where does the

\frac{1}{2}

chance of each gamete allele come from?

Mendel's Law of Segregation — ek heterozygote ke dono alleles gametes mein equally separate ho jaate hain.

What probability factor corresponds to a heterozygous offspring at one gene?

\frac{1}{2}

(

1 : 2 : 1

genotype ratio ka middle term).

Connections

Law of Segregation — har gamete mein ½ probability deta hai.
Law of Independent Assortment — genes ke across multiply karna justify karta hai.
Punnett Squares — inhi calculations ka visual version.
Dihybrid Cross 9-3-3-1 — do monohybrid ratios ka direct product.
Binomial Probability in Genetics — "exactly k of n offspring" ke liye.
Linkage and Recombination — jahan product rule fail ho jaata hai.
Test Cross — unknown genotype infer karne ke liye probability use karna.