Prove or disprove. Given n (n > 1) events A. A,. .A, E5, then n PIA]- PIA, N A,] < PIJ4;]< P[A,] i=1
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A: The solution is given below in the next step:
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A: Definition of Mutually exclusive events
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A: We have, B B' Total A 96 40 A' 69 45 Total
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- an ice-cream stall m + n (n ≤ m) people are in a queue to buy one |50 ice-cream each. The stall-keeper has no change. Precisely m of the people have just a |50 note and the remaining n people have just a |100 note. Find the probability that the stall-keeper always has enough change to serve the customers in turn.Two numbers are randomly picked from the ranges [-1,2] and [1,3]. a) Write the sample space.b) What is the probability that the sum of these two numbers is less than 4 but greater than 2?c) What is the probability that the sum of these two numbers equals 4?d) The sum of these two numbers is less than 4, greater than 2 and x is greater than yWhat are the chances of it happening?just using the definition of probability and the theorem on page 17 of the notes, prove thatfor any events A and B,P (A) + P (B) − 1 ≤ P (A ∩ B) ≤ P (A ∪ B) ≤ P (A) + (B).(Please label them as “Inquality I”, “First In-equality”, “Inequality A”, etc.)Note: You may use Venn diagrams as guides, but a picture is nota complete proof. give proofs in complete sentences and mathematicalwork
- a) Determine the probability that a randomly selected call was during the week. The probability is----- (Round to three decimal places as needed.) b. Determine the probability that a randomly selected call was less than 20 minutes. The probability is----- (Round to three decimal places as needed.) c. Determine the probability that a randomly selected call was 10 to less than 20 minutes. The probability is----- (Round to three decimal places as needed.) A call center for customer support records the time each customer requires for service and whether the call occurred during the week (Monday through Friday) or during the weekend (Saturday and Sunday). The results are shown in the accompanying table. Complete parts a through h. Time (minutes) Week Weekend Less than 5 15 4 5 to less than 10 27 3 10 to less than 15 12 8 15 to less than 20 23 5 More than 20 10 14b) Events A and B are such that P(A) = 3/5, P(B) = 4/15. Find P(A U B) if i) A and B are mutually exclusive events. ii) A and B are independent events.The probability that a girl chosen at random is a left-hander is 2/7. Calculate the probability that among a group of ten girls chosen at random...