Probability
Independent Events and Complements
Grade 12
Question:
<p>Let n ordinary fair dice are rolled once. The probability that at least one of the dice shows an odd number is \(\frac{31}{32}\) than 'n' is equal to:</p>
<p>(a) 3</p>
<p>(b) 4</p>
<p>(c) 5</p>
<p>(d) 6</p>
Step-by-Step Solution
Key Concept: Use complementary probability: P(at least one) = 1 - P(none), and recognize that 1/32 = (1/2)^5.
<p>Probability of getting an odd number on a single die = $\frac{3}{6} = \frac{1}{2}$</p><p>Probability of getting an even number on a single die = $\frac{1}{2}$</p><p>Probability that all n dice show even numbers = $\left(\frac{1}{2}\right)^n$</p><p>Probability that at least one die shows an odd number = $1 - \left(\frac{1}{2}\right)^n$</p><p>Given: $1 - \left(\frac{1}{2}\right)^n = \frac{31}{32}$</p><p>$\left(\frac{1}{2}\right)^n = 1 - \frac{31}{32} = \frac{1}{32}$</p><p>$\left(\frac{1}{2}\right)^n = \frac{1}{2^5}$</p><p>Therefore: $n = 5$</p>
Correct Answer: d