Probability
Classical Probability
Grade 12
Question:
<p>The probability that a positive integral power of 137 will have the digit 7 in units place of the value of the power, is</p>
<p>A. \(\dfrac{1}{3}\)</p>
<p>B. \(\dfrac{1}{2}\)</p>
<p>C. \(\dfrac{1}{4}\)</p>
<p>D. \(\dfrac{3}{4}\)</p>
Step-by-Step Solution
Key Concept: Find the pattern of units digits for successive powers of 137 by observing the cycle of units digits of powers of 7 (since only the units digit of the base affects the units digit of the result).
<p><strong>Step 1:</strong> Identify the units digit pattern. Since 137 has units digit 7, we need to find the pattern of units digits for powers of 7.</p><p><strong>Step 2:</strong> Calculate successive powers of 7 (mod 10):</p><ul><li>7¹ ≡ 7 (mod 10)</li><li>7² = 49 ≡ 9 (mod 10)</li><li>7³ = 343 ≡ 3 (mod 10)</li><li>7⁴ = 2401 ≡ 1 (mod 10)</li><li>7⁵ ≡ 7 (mod 10) [cycle repeats]</li></ul><p><strong>Step 3:</strong> The units digits follow the cycle: {7, 9, 3, 1} with period 4.</p><p><strong>Step 4:</strong> Out of every 4 consecutive powers, exactly 1 power (when exponent ≡ 1 mod 4) has units digit 7.</p><p><strong>Step 5:</strong> For a randomly chosen positive integer power n, the probability that 137ⁿ has units digit 7 is:</p><p>P = (number of favorable outcomes in one cycle)/(cycle length) = 1/4</p><p>∴ Answer: C (assuming option C is 1/4)</p>
Correct Answer: C