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Give step-by-step solution with explanation and final answer:Problem 1: The chart below shows the number of ice cream cones sold at Buddy's Ice Cream Parlor £rom Monday through Friday: [ [wonday [Tuesday [Wednesday [Thursday [Friday | Fri fee Lis De Le fw fo | A computer commiting firm presently bas bids out sm three projects. Lat A= [Oosstas Juz Jor fot Iw Jo | (awarded project 0), for 1 = 13,3. Swpese that: P(A) = 0.22, PU = 0.25, (EE EEN ECT ER ETN EC PAD=028 . PANAD=011 . PUAAD=005 . PUA) =0.07, PUAL(143 142) = 0.01. Express in words each of the following events, and compute a Wat vas the mess sale for vanilla? To hair of aN ses 5. That vas the mean sale for all flavors on Nonday? ©. Find the nedian datly sales for chocolate. . aUA 4 Construct histogram for strasberry ice cream cones sold. wana © AUA UA Problen 2: 4 Anna Blood pressure values are often reported to the nearest 5 malig (100, 105, 110, ete.), ANAM, suppose the actual blood pressure values for nine randomly selected individuals are: 1 (ANADUA, LBS 124 134 10.0 LNT 120 1083 135 188.2 Problem 6: oho main oi reparti Bisod sass; viiieid Prove the proposition of span: PAU BLIC) fn page 63 of the textbook. b. That 1s the mean of the reported blood pressure values? What fs the 108 trimmed sean? ©. Suppose the blood pressure of the second individual is 127.6 rather than 127.4, How does this affect the median, mean, and 10% trimmed mean of the reported PROPOSITION | For any three events A. B, andl C. values? That docs this say about the sensitivity of these data? PAUBUO = PA) + AB) + RO) ~ PAB) ~ PA NC) “PEOC+ AANENG Probes 3: Use Venn diagrams to verify the De Morgan’ 3 law: “This can be verified by examining the Venn diagram of A U # UC. shown in Figure . UUW ane 26 When PU), AB), snd LC) ar add. the ection ecb PUA 1). A 0 C.and AUR 11 ell coun twice. Ech one at thereon be sbencied. bans =aus But hen UA (110) hs bo aod in the ie ed sured ot de dims, 201t must be addd bck. In gener, the probability of + union of £ vos i bind Problem 4: ‘by summing individual event probabilities. subtracting double intersection probabilities. A family constating of three persons, A B, and C, goes to a medical clinic that oro se sl always Bas a doctor at each of stations 1,2, aad 3. During a certain week, each ES mesber of the family visits the clinic once and is assigned at random to a station. r \¥ The copertant conto of recat the tation mabe fo cach member On otc iid 15 (1,2,1) for A to station 1, B to station 2, C to station L. [rp a List the 27 outcomes fn the sample space. b. List all cutcoses in the event that all three mesbers go to the same station. ©. List all outcomes in the event that all mesbers o to different station. & List all cutcones n the event that 50 one goes to station 2.

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Give step-by-step solution with explanation and final answer:Uploaded ImageProblem 1: The chart below shows the number of ice cream cones sold at Buddy's Ice Cream Parlor £rom Monday through Friday: [ [wonday [Tuesday [Wednesday [Thursday [Friday | Fri fee Lis De Le fw fo | A computer commiting firm presently bas bids out sm three projects. Lat A= [Oosstas Juz Jor fot Iw Jo | (awarded project 0), for 1 = 13,3. Swpese that: P(A) = 0.22, PU = 0.25, (EE EEN ECT ER ETN EC PAD=028 . PANAD=011 . PUAAD=005 . PUA) =0.07, PUAL(143 142) = 0.01. Express in words each of the following events, and compute a Wat vas the mess sale for vanilla? To hair of aN ses 5. That vas the mean sale for all flavors on Nonday? ©. Find the nedian datly sales for chocolate. . aUA 4 Construct histogram for strasberry ice cream cones sold. wana © AUA UA Problen 2: 4 Anna Blood pressure values are often reported to the nearest 5 malig (100, 105, 110, ete.), ANAM, suppose the actual blood pressure values for nine randomly selected individuals are: 1 (ANADUA, LBS 124 134 10.0 LNT 120 1083 135 188.2 Problem 6: oho main oi reparti Bisod sass; viiieid Prove the proposition of span: PAU BLIC) fn page 63 of the textbook. b. That 1s the mean of the reported blood pressure values? What fs the 108 trimmed sean? ©. Suppose the blood pressure of the second individual is 127.6 rather than 127.4, How does this affect the median, mean, and 10% trimmed mean of the reported PROPOSITION | For any three events A. B, andl C. values? That docs this say about the sensitivity of these data? PAUBUO = PA) + AB) + RO) ~ PAB) ~ PA NC) “PEOC+ AANENG Probes 3: Use Venn diagrams to verify the De Morgan’ 3 law: “This can be verified by examining the Venn diagram of A U # UC. shown in Figure . UUW ane 26 When PU), AB), snd LC) ar add. the ection ecb PUA 1). A 0 C.and AUR 11 ell coun twice. Ech one at thereon be sbencied. bans =aus But hen UA (110) hs bo aod in the ie ed sured ot de dims, 201t must be addd bck. In gener, the probability of + union of £ vos i bind Problem 4: ‘by summing individual event probabilities. subtracting double intersection probabilities. A family constating of three persons, A B, and C, goes to a medical clinic that oro se sl always Bas a doctor at each of stations 1,2, aad 3. During a certain week, each ES mesber of the family visits the clinic once and is assigned at random to a station. r \¥ The copertant conto of recat the tation mabe fo cach member On otc iid 15 (1,2,1) for A to station 1, B to station 2, C to station L. [rp a List the 27 outcomes fn the sample space. b. List all cutcoses in the event that all three mesbers go to the same station. ©. List all outcomes in the event that all mesbers o to different station. & List all cutcones n the event that 50 one goes to station 2.

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Asked by: Sahzadi
Created at: 2026-03-07 21:14:44

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Let's solve each problem step by step. --- # **Problem 1** ## *Given Table:* | Flavor | Monday | Tuesday | Wednesday | Thursday | Friday | |------------|--------|---------|-----------|----------|--------| | Vanilla | 15 | 17 | 20 | 18 | 17 | | Chocolate | 22 | 25 | 18 | 14 | 12 | | Strawberry | 12 | 13 | 24 | 18 | 17 | ### **a. What was the mean sale for vanilla?** \[ \text{Mean} = \frac{15 + 17 + 20 + 18 + 17}{5} = \frac{87}{5} = 17.4 \] ### **b. What was the mean sale for all flavors on Monday?** \[ \text{Total Monday} = 15 + 22 + 12 = 49 \] \[ \text{Mean (Monday)} = \frac{49}{3} = 16.33 \] ### **c. Find the median daily sales for chocolate.** Chocolate sales: 22, 25, 18, 14, 12 (Sort: 12, 14, 18, 22, 25) Median = 18 (middle value) ### **d. Construct a histogram for strawberry ice cream sales.** **Strawberry Sales:** 12, 13, 24, 18, 17 - 12-14: 2 days (12, 13) - 15-19: 2 days (18, 17) - 20-24: 1 day (24) (Histogram would be bars: 2, 2, 1 for the ranges above.) --- # **Problem 2** Blood pressures: **118.6, 127.4, 138.4, 130., 113.7, 122., 108.3, 131.5, 133.2** ### **a. Median of reported blood pressures** Sort: 108.3, 113.7, 118.6, 122., 127.4, 130., 131.5, 133.2, 138.4 **Median** = 127.4 (5th value in a sorted list of 9) ### **b. Mean of reported blood pressures** \[ \text{Mean} = \frac{108.3 + 113.7 + 118.6 + 122. + 127.4 + 130. + 131.5 + 133.2 + 138.4}{9} = \frac{1123.1}{9} = 124.79 \] ### **c. 10% trimmed mean** 10% of 9 = .9 ≈ 1 (trim lowest and highest value) Remove: 108.3, 138.4 Values left: 113.7, 118.6, 122., 127.4, 130., 131.5, 133.2 \[ \text{Trimmed Mean} = \frac{113.7 + 118.6 + 122. + 127.4 + 130. + 131.5 + 133.2}{7} = \frac{876.4}{7} = 125.20 \] ### **d. If 127.4 is changed to 127.6** New sum: 1123.1 + .2 = 1123.3 New mean: 1123.3 / 9 = 124.81 (change is very small) Median changes from 127.4 to 127.6 Trimmed mean: Replace 127.4 with 127.6 in trimmed mean, so \[ \frac{876.4 + .2}{7} = 876.6 / 7 = 125.23 \] **Conclusion:** Changing a single value by a small amount does not significantly affect mean or trimmed mean, but median will change if the value was the median. --- # **Problem 3** ## **De Morgan’s Law Verification** a. \((A \cup B)' = A' \cap B'\) b. \((A \cap B)' = A' \cup B'\) **Use Venn diagrams**: - Shade \(A\) and \(B\), then shade the outside for complement. - Show that the outside region is the intersection of outside \(A\) and outside \(B\). --- # **Problem 4** **Sample Space:** Each person (A, B, C) can go to stations 1, 2, or 3. Total outcomes: \(3 \times 3 \times 3 = 27\) Let each outcome be (a, b, c) where a is A's station, etc. ### **a. List the 27 outcomes** (1,1,1), (1,1,2), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,1), (1,3,2), (1,3,3), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,2,2), (2,2,3), (2,3,1), (2,3,2), (2,3,3), (3,1,1), (3,1,2), (3,1,3), (3,2,1), (3,2,2), (3,2,3), (3,3,1), (3,3,2), (3,3,3) ### **b. All go to same station** (1,1,1), (2,2,2), (3,3,3) ### **c. All go to different stations** Possible permutations of (1,2,3): (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) ### **d. No one goes to station 2** Each person can go to 1 or 3 only. Possible outcomes: (1,1,1), (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,1,3), (3,3,1), (3,3,3) --- # **Problem 5** Given probabilities for three projects \(A_1, A_2, A_3\): - \(P(A_1) = .22\), \(P(A_2) = .25\), \(P(A_3) = .28\) - \(P(A_1 \cap A_2) = .11\), \(P(A_1 \cap A_3) = .05\), \(P(A_2 \cap A_3) = .07\) - \(P(A_1 \cap A_2 \cap A_3) = .01\) Compute probabilities for: a. \(A_1 \cup A_2\): \[ P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2) = .22 + .25 - .11 = .36 \] b. \(A_1 \cap A_2\): \[ P(A_1 \cap A_2) = .11 \] c. \(A_1 \cup A_2 \cup A_3\): \[ \begin{align*} P(A_1 \cup A_2 \cup A_3) &= P(A_1) + P(A_2) + P(A_3) \\ &- P(A_1 \cap A_2) - P(A_1 \cap A_3) - P(A_2 \cap A_3) \\ &+ P(A_1 \cap A_2 \cap A_3) \\ = .22 + .25 + .28 - .11 - .05 - .07 + .01 = .53 \end{align*} \] d. \(A_1 \cap (A_2 \cup A_3)\): \[ P(A_1 \cap (A_2 \cup A_3)) = P((A_1 \cap A_2) \cup (A_1 \cap A_3)) \] \[ = P(A_1 \cap A_2) + P(A_1 \cap A_3) - P(A_1 \cap A_2 \cap A_3) \] \[ = .11 + .05 - .01 = .15 \] e. \((A_1 \cap A_2) \cup A_3\) \[ P((A_1 \cap A_2) \cup A_3) = P(A_1 \cap A_2) + P(A_3) - P((A_1 \cap A_2) \cap A_3) \] But \((A_1 \cap A_2) \cap A_3 = A_1 \cap A_2 \cap A_3\) \[ = .11 + .28 - .01 = .38 \] --- # **Problem 6** **Prove:** \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \] **This is the inclusion-exclusion principle**, already shown in the blue box in your image. --- ## **Final Answers Summary** **1a:** 17.4 **1b:** 16.33 **1c:** 18 **1d:** Histogram: [2 (12-14), 2 (15-19), 1 (20-24)] **2a:** 127.4 **2b:** 124.79 **2c:** 125.20 **2d:** Median: 127.6, Mean: 124.81, Trimmed mean: 125.23 (small changes) **3:** Use Venn diagrams as described **4a:** 27 listed outcomes **4b:** (1,1,1), (2,2,2), (3,3,3) **4c:** (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) **4d:** (1,1,1), (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,1,3), (3,3,1), (3,3,3) **5a:** .36 **5b:** .11 **5c:** .53 **5d:** .15 **5e:** .38 **6:** Inclusion-Exclusion Principle shown If you need a detailed solution for any specific part, let me know!

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