Probability
Conditional Probability and Total Probability
Grade 12
Question:
<p>The probability that in a year of 22nd century chosen at random, there will be 53 Sundays, is</p>
<p>(a) \(\frac{3}{28}\)</p>
<p>(b) \(\frac{2}{28}\)</p>
<p>(c) \(\frac{7}{28}\)</p>
<p>(d) \(\frac{5}{28}\)</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on leap and non-leap years, which have different numbers of Sundays in a year.
<p><strong>Solution:</strong> In the 22nd century, there are 25 leap years (2100, 2104, 2108, ..., 2196) and 75 non-leap years.</p><p>Let $E_1$ = Selecting a leap year from 22nd century</p><p>Let $E_2$ = Selecting a non-leap year from 22nd century</p><p>Let $E$ = There are 53 Sundays in a year of 22nd century</p><p>We have $P(E_1) = \frac{25}{100}$, $P(E_2) = \frac{75}{100}$, $P\left(\frac{E}{E_1}\right) = \frac{2}{7}$, and $P\left(\frac{E}{E_2}\right) = \frac{1}{7}$</p><p>Required probability: $P(E) = P((E \cap E_1) \cup (E \cap E_2)) = P(E \cap E_1) + P(E \cap E_2) = P(E_1) \cdot P\left(\frac{E}{E_1}\right) + P(E_2) \cdot P\left(\frac{E}{E_2}\right)$</p><p>$= \frac{25}{100} \cdot \frac{2}{7} + \frac{75}{100} \cdot \frac{1}{7} = \frac{50}{700} + \frac{75}{700} = \frac{125}{700} = \frac{5}{28}$</p>
Correct Answer: d