Three students , , and are given a problem to solve. Consider the following events:: At least one of , , and can solve the problem,: can solve the problem, given that neither nor can solve the problem,: can solve the problem and cannot solve the problem,: can solve the problem.For any event , let denote the probability of . Ifand where and are coprime positive integers, find the value of .\
[THOUGHT] Define the events properly. Let x = P(none solve) = P(¬S1 ∧ ¬S2 ∧ ¬S3).
From P(B) = P(S1 | ¬S2 ∧ ¬S3) = 1/2, it follows that P(only S1) = x, so P(¬S2 ∧ ¬S3) = 2x.
P(C) = P(S2 ∧ ¬S3) = 1/10.
P(¬S3) = P(¬S2 ∧ ¬S3) + P(S2 ∧ ¬S3) = 2x + 1/10.
But P(¬S3) = 1 - P(D) = 1 - 1/12 = 11/12.
Thus, 2x + 1/10 = 11/12.
Solve: 2x = 11/12 - 1/10 = (110 - 12)/120 = 98/120 = 49/60.
x = 49/120.
P(A) = 1 - x = 71/120.
p=71, q=120 coprime.
Verify with Python: compute 11/12 - 1/10, etc., and p² + q². [/THOUGHT] [PYTHON] from fractions import Fraction
p_not_s3 = Fraction(11,12) p_c = Fraction(1,10) p_not_s2_not_s3 = Fraction(2,1) * Fraction(49,120) # but solve properly
diff = Fraction(11,12) - Fraction(1,10) print("11/12 - 1/10 =", diff)
x = diff / 2 print("x =", x)
p_a = 1 - x print("P(A) =", p_a)
p, q = p_a.numerator, p_a.denominator print("p =", p, "q =", q)
result = p2 + q2 print("p^2 + q^2 =", result) [/PYTHON]
71² + 120² = 5041 + 14400 = 19441
[THOUGHT] The Python output confirms P(A) = 71/120, so p=71, q=120, and p² + q² = 19441. [/THOUGHT] \boxed{19441}
Explanation
The reasoning begins by defining key probabilities based on the given events. Let x represent P(¬S1 ∧ ¬S2 ∧ ¬S3), the probability that none of the students solve the problem.
From P(B) = P(S1 | ¬S2 ∧ ¬S3) = 1/2, it follows that P(S1 ∧ ¬S2 ∧ ¬S3) = (1/2) × P(¬S2 ∧ ¬S3), so P(¬S2 ∧ ¬S3) = 2x.
P(C) = P(S2 ∧ ¬S3) = 1/10.
P(¬S3) = P(¬S2 ∧ ¬S3) + P(S2 ∧ ¬S3) = 2x + 1/10.
Since P(D) = P(S3) = 1/12, then P(¬S3) = 1 - 1/12 = 11/12.
Set up the equation: 2x + 1/10 = 11/12.
Solve for x: 2x = 11/12 - 1/10 = (110 - 12)/120 = 98/120 = 49/60, so x = (49/60) / 2 = 49/120.
Thus, P(A) = 1 - x = 1 - 49/120 = 71/120.
Here, 71 and 120 are coprime, so p = 71 and q = 120.
Finally, compute p² + q² = 71² + 120² = 5041 + 14400 = 19441, as verified by the Python output.