Probability — Important Questions

3/4

Step 1: Sample space when two coins are tossed: S = {HH, HT, TH, TT}, so n(S) = 4. Step 2: 'At most one head' means 0 heads OR 1 head. Step 3: Event with 0 heads = {TT} → 1 outcome. Event with exactly 1 head = {HT, TH} → 2 outcomes. Total favorable = 1 + 2 = 3. Step 4: P(at most one head) = 3/4. Step 5: Common mistake: Students often confuse 'at most one' with 'exactly one'. 'At most one' means ≤ 1, so it includes both 0 heads and 1 head. Alternatively, P(at most one head) = 1 - P(two heads) = 1 - 1/4 = 3/4 using complementary approach.

2/13

Step 1: A standard deck has 52 cards with 4 suits. Each suit has 13 cards including one King and one Queen. Step 2: Total Kings in deck = 4, Total Queens in deck = 4. Step 3: Event E = drawing a King OR a Queen. Since a card cannot be both a King and a Queen simultaneously, these are mutually exclusive events. Step 4: n(E) = 4 + 4 = 8 favorable outcomes. P(E) = 8/52 = 2/13. Step 5: Note that 8/52 and 2/13 are the SAME value (simplified). Common mistake is writing 4/52 = 1/13, which only counts kings OR only queens, not both.

1/10

Step 1: We need numbers from 1 to 30 divisible by BOTH 4 and 6. A number divisible by both 4 and 6 must be divisible by LCM(4, 6). Step 2: LCM(4, 6) = 12. So we need multiples of 12 from 1 to 30. Step 3: Multiples of 12 up to 30: 12, 24 → only 2 numbers. Step 4: P = 2/30 = 1/15. Wait — let me recount: 12 × 1 = 12, 12 × 2 = 24, 12 × 3 = 36 > 30. So favorable outcomes = {12, 24}, n(E) = 3. Correction: Actually only 2 multiples. P = 2/30 = 1/15. Step 5: The correct answer based on option matching: {12, 24} gives 2/30 = 1/15. However, the listed correct option is 1/10 which would mean 3 favorable.

2/5

Step 1: Total people = 200, Tea = 120, Coffee = 90, Both = 40. Step 2: People who like tea but NOT coffee = (People who like tea) - (People who like both) = 120 - 40 = 80. Step 3: P(tea but not coffee) = 80/200 = 2/5. Step 4: Verify: 2/5 = 0.4. This means 40% of people like only tea. Step 5: Common mistake: Students sometimes use 120/200 = 3/5 which gives probability for ALL tea drinkers including those who also like coffee. The key word 'but NOT coffee' means we must subtract the overlap.