34. Mendel crossed tall pea plants with dwarf ones. The F₁ plants were all tall. When these F₁ plants were selfed to produce F₂ generation, he got a 3: 1 tall to dwarf ratio in the offspring. What is the probability that out of three plants (of F₂ generation) picked up at random two would be dwarf and one would be tall?
(1) 3/4          (2) 3/8
(3) 9/64        (4) 9/32

Step-by-step reasoning

From the monohybrid cross

In F₂: P(tall) = 3/4, P(dwarf) = 1/4

Let dwarf = “success” with probability p = 1/4, tall = “failure” with q = 3/4 [web:89][web:92].

Binomial probability setup

We choose 3 plants, wanting exactly 2 dwarf and 1 tall.

Binomial probability formula:

de>P(exactly k dwarfs in n plants) = (n k) p^k q^n-k

Here n = 3, k = 2, p = 1/4, q = 3/4:

Calculation

de>P = (3 2) (1/4)² (3/4)¹ = 3 × (1/16) × (3/4) = 3 × 3/64 = 9/64

So the probability is 9/64 [web:90].

Why other options are incorrect

• 3/4: probability that a single F₂ plant is tall
• 3/8: assumes 1/2 tall and 1/2 dwarf (wrong 3:1 ratio)
• 9/32: equals 2 × 9/64; incorrect permutation count

Therefore, the probability that among three randomly picked F₂ plants two are dwarf and one is tall is 9/64.