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PEAMCQModerate Needs review

2023 PEA Q23

Arjun and Gukesh each toss 3 fair coins. p1=P(Arjun’s heads>Gukesh’s heads)p_1=P(\text{Arjun's heads}>\text{Gukesh's heads}). Find p1p_1.

The available source text does not include a full option block for this item, so the question is marked needsReview.
Reveal answer and solution

Answer

C

Solution

  1. 1

    Let A,G∼Bin(3,1/2)A,G\sim\mathrm{Bin}(3,1/2) independent. By symmetry, P(A>G)=P(G>A)P(A>G)=P(G>A). Let q=P(A=G)q=P(A=G):

  2. 2
    2P(A>G)+q=1  ⟹  P(A>G)=1−q2. 2 P(A>G)+q=1 \;\Longrightarrow\; P(A>G)=\frac{1-q}{2}.
  3. 3
    q=∑k=03[(3k)2−3]2=1+9+9+164=2064=516. q=\sum_{k=0}^{3}\bigl[\tbinom{3}{k}2^{-3}\bigr]^2=\frac{1+9+9+1}{64}=\frac{20}{64}=\frac{5}{16}.
  4. 4
    p1=1−5/162=11/162=1132. p_1=\frac{1-5/16}{2}=\frac{11/16}{2}=\frac{11}{32}.

Answer structure / marking notes

Needs review: source TeX does not provide a full four-option MCQ block.

Content note

Imported from public/resources/isi/msqe/solutions/pea/2023/ISI_MSQE_PEA_2023_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.