Probability
Binomial Distribution
Grade 12
Question:
<p>There is 30% chance that it rains on any particular day. Then in a period of 7 days, which of the following is/are correct?</p><p>(a) The probability that there is at least one rainy day within a period of 7 days is \(1-\left(\dfrac{7}{10}\right)^6\)</p><p>(b) The probability that there is at least one rainy day within a period of 7 days is \(1-\left(\dfrac{7}{10}\right)^7\)</p><p>(c) Given that there is at least one rainy day, what is the probability that there are at least two rainy days is \(\dfrac{1-\left(\dfrac{7}{10}\right)^7 - 7\left(\dfrac{3}{10}\right)\left(\dfrac{7}{10}\right)^7}{1-\left(\dfrac{7}{10}\right)^7}\)</p><p>(d) Given that there is at least one rainy day, what is the probability that there are at least two rainy days is \(\dfrac{1-\left(\dfrac{7}{10}\right)^7 - 7\left(\dfrac{3}{10}\right)\left(\dfrac{7}{10}\right)^6}{1-\left(\dfrac{7}{10}\right)^7}\)</p>
<p>(a) The probability that there is at least one rainy day within a period of 7 days is \(1-\left(\dfrac{7}{10}\right)^6\)</p>
<p>(b) The probability that there is at least one rainy day within a period of 7 days is \(1-\left(\dfrac{7}{10}\right)^7\)</p>
<p>(c) Given that there is at least one rainy day, the probability that there are at least two rainy days is \(\dfrac{1-\left(\dfrac{7}{10}\right)^7 - 7\left(\dfrac{3}{10}\right)\left(\dfrac{7}{10}\right)^7}{1-\left(\dfrac{7}{10}\right)^7}\)</p>
<p>(d) Given that there is at least one rainy day, the probability that there are at least two rainy days is \(\dfrac{1-\left(\dfrac{7}{10}\right)^7 - 7\left(\dfrac{3}{10}\right)\left(\dfrac{7}{10}\right)^6}{1-\left(\dfrac{7}{10}\right)^7}\)</p>
Step-by-Step Solution
Key Concept: Use complement rule for 'at least one' events: P(at least 1 rain) = 1 - P(no rain in all 7 days). For conditional probability, apply: P(A|B) = P(A∩B)/P(B), where A = 'at least 2 rainy days' and B = 'at least 1 rainy day'.
<p><strong>Step 1: Probability of at least one rainy day</strong></p><p>P(no rain in 7 days) = (0.7)^7 = (7/10)^7</p><p>P(at least 1 rainy day) = 1 - (7/10)^7 ✓ <strong>Option (b) is correct</strong></p><p></p><p><strong>Step 2: Conditional probability - at least 2 rainy days given at least 1</strong></p><p>Let A = at least 2 rainy days, B = at least 1 rainy day</p><p>P(A|B) = P(A∩B)/P(B) = P(A)/P(B) [since A ⊂ B]</p><p></p><p><strong>Step 3: Calculate P(at least 2 rainy days)</strong></p><p>P(at least 2) = 1 - P(0 rainy days) - P(exactly 1 rainy day)</p><p>= 1 - (7/10)^7 - C(7,1)(3/10)(7/10)^6</p><p>= 1 - (7/10)^7 - 7(3/10)(7/10)^6</p><p></p><p><strong>Step 4: Apply conditional probability formula</strong></p><p>P(at least 2 | at least 1) = [1 - (7/10)^7 - 7(3/10)(7/10)^6] / [1 - (7/10)^7]</p><p>✓ <strong>Option (d) is correct</strong></p><p></p><p><strong>Why (a) and (c) are wrong:</strong></p><p>(a) Uses (7/10)^6 instead of (7/10)^7 - incorrect exponent</p><p>(c) Uses (7/10)^7 instead of (7/10)^6 in the subtracted term - confuses binomial coefficient calculation</p><p></p><p>∴ Answer: <strong>B, D</strong></p>
Correct Answer: B,D