1. **Problem 02: Probability with card selections** (a) Find the probability the second card is a face card given the first card was red. - A standard deck has 52 cards: 26 red, 26 black. - Number of face cards: 12 (Jack, Queen, King of each suit). Half are red (6) and half are black (6). - Since the first card is red and replaced, the deck composition remains unchanged for second draw. - Probability(second card is face card | first card red) = Probability(second card is face card) = $\frac{12}{52} = \frac{3}{13} \approx 0.2308$. (b) Find the probability second card is an ace given the first was a face card. - 4 aces in the deck. Since replacement, deck unchanged. - Probability(second card is ace | first card face card) = Probability(second card is ace) = $\frac{4}{52} = \frac{1}{13} \approx 0.0769$. (c) Find the probability second card is a black jack given the first card was a red ace. - There are 2 black jacks in a deck (Jack of spades and Jack of clubs). - First card red ace replaced, deck unchanged. - Probability(second card is black jack | first red ace) = Probability(second card is black jack) = $\frac{2}{52} = \frac{1}{26} \approx 0.0385$. 2. **Problem 03: Probability from prison escape data** - Total seasons each: 45 (Winter, Spring, Summer, Fall). (a) Probability escapes between 16 and 20 in winter: - Escapes between 16 and 20 in winter = 3 seasons. - Total winter seasons = 45. - Probability = $\frac{3}{45} = \frac{1}{15} \approx 0.0667$. (b) Probability more than 10 escapes attempted in summer: - Categories >10: 11–15 (7), 16–20 (6), 21–25 (5), More than 25 (4). - Sum = 7 + 6 + 5 + 4 = 22. - Total summer seasons = 45. - Probability = $\frac{22}{45} \approx 0.4889$. (c) Probability escapes between 11 and 20 in any season (all seasons combined): - Categories combined: 11–15 + 16–20. - Sum for each season: - Winter: 5 + 3 = 8 - Spring: 8 + 4 = 12 - Summer: 7 + 6 = 13 - Fall: 7 + 5 = 12 - Total combined = 8 + 12 + 13 + 12 = 45. - Total seasons = 45 * 4 = 180. - Probability = $\frac{45}{180} = \frac{1}{4} = 0.25$. 3. **Problem 04: Probability from socio-economic survey** - Total respondents = 1000. (a) Probability respondent is Undergraduate (U): - Total U = 300. - Probability = $\frac{300}{1000} = 0.3$. (b) Probability respondent is Graduate (G): - Total G = 700. - Probability = $\frac{700}{1000} = 0.7$. (c) Probability respondent is Female (F): - Total F = 400. - Probability = $\frac{400}{1000} = 0.4$. (d) Probability Male-Graduate (MG): - MG = 450. - Probability = $\frac{450}{1000} = 0.45$. (e) Probability Undergraduate-Female (UF): - UF = 150. - Probability = $\frac{150}{1000} = 0.15$. (f) Probability Graduate given Female (G|F): - Graduate Females = 250, Total females = 400. - Probability = $\frac{250}{400} = 0.625$. (g) Probability Male given Undergraduate (M|U): - Male Undergraduates = 150, Total undergraduates = 300. - Probability = $\frac{150}{300} = 0.5$. 4. **Problem 05: Produce shipper fruit conditions** - Total boxes = 10,000. - Ecuador boxes = 6000, Honduran boxes = 4000. (a) Probability box has damaged fruit: - Damaged total = 200 + 365 = 565. - Probability = $\frac{565}{10000} = 0.0565$. Probability box has overripe fruit: - Overripe total = 840 + 295 = 1135. - Probability = $\frac{1135}{10000} = 0.1135$. (b) Probability box is from Ecuador or Honduras: - All boxes are from either Ecuador or Honduras. - Probability = 1. (c) Probability box is from Honduras given it contains overripe fruit: - Overripe from Honduras = 295. - Overripe total = 1135. - Probability = $\frac{295}{1135} \approx 0.2599$. (d) Probability box has damaged or overripe fruit if mutually exclusive: - Sum of damaged + overripe = 565 + 1135 = 1700. - Probability = $\frac{1700}{10000} = 0.17$. If not mutually exclusive, assume no overlap given. Thus same result of 0.17. **Final answers:** - Problem 02: (a) $\frac{3}{13}$, (b) $\frac{1}{13}$, (c) $\frac{1}{26}$. - Problem 03: (a) $\frac{1}{15}$, (b) $\frac{22}{45}$, (c) $\frac{1}{4}$. - Problem 04: (a) 0.3, (b) 0.7, (c) 0.4, (d) 0.45, (e) 0.15, (f) 0.625, (g) 0.5. - Problem 05: (a) damaged 0.0565, overripe 0.1135, (b) 1, (c) 0.2599, (d) 0.17 if mutually exclusive or not.