Probability
Probability with product of outcomes
nta_pyq_2023_jan
Grade 12
Question:
If an unbiased die, marked with $-2, -1, 0, 1, 2, 3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:
$\frac{881}{2592}$
$\frac{521}{2592}$
$\frac{440}{2592}$
$\frac{27}{288}$
Step-by-Step Solution
Key Concept: Product is positive when all 5 outcomes are positive OR exactly 2 or 4 are negative (with no zeros); handle zero separately
Positive faces: $\{1,2,3\}$ (3 faces), negative: $\{-2,-1\}$ (2 faces), zero: $\{0\}$ (1 face). Product positive iff no zeros and even number of negatives. $P(\text{positive}) = {}^5C_5(1/2)^5 + {}^5C_2(1/3)^2(1/2)^3 + {}^5C_4(1/3)^4(1/2)^1 = 1/32 + 10\times1/9\times1/8 + 5\times1/81\times1/2 = 521/2592$. Answer: (2)
Correct Answer: $\frac{521}{2592}$