MATH221 Mathematics for Computer ScienceTutorial Sheet Week 13 – Autumn 20211. The two-way table below gives the thousands of commuters in Massachusetts in 2015 by transportationmethod and one-way commute length.(a) Given that a randomly selected commuter used public transportation, find the probability that he had acommute of 60 minutes or more.(b) Given that a randomly selected commuter had a commute of 60 minutes or more, find the probability thathe used public transportation.2. You toss a fair coin three times.(a) What is the probability that you observe exactly two tails?(b) Given that you have observed at least one tail, what is the probability that you observed at least two tails?3. Business owners and partners Yousef and Olivia sold me a product two years ago, the Yousef{Olivia Super{Happy{Important{Thing. This product is known to function properly until it breaks down after t years withprobabilityP(T ≥ t) = e- 5t :For example, the probability that the product lasts 1 year or more isP(T ≥ 1) = e- 1 5 ≈ 0:82:Since I have used my Super{Happy{Important{Thing for the past two years without incident, how likely is it thatit will stop working in the coming year?4. You are a respectable basketball player with a 70% chance of making a free throw at any time. You are at acharity event that involves a competition; the most free throws out of 5 attempts wins. You know that the othercompetitors are average players and will probably score less than your most likely score.(a) Graph a histogram of all your possible outcomes and another of all cumulative outcomes.(b) Determine your most probable minimum performance necessary to win the competition and the probabilitythat this will happen.5. Yousef produces 20 Super{Happy{Important{Things per day, with a 2% defective rate. Olivia is 50% fasterthan her partner, but she produces double the rate of defective items. At the end of every day, the products arepacked randomly into one box of 20 items and one box of 30 items.Today, the owner of the umbrella corporation Super{Happy{Important{Things{R{Us, Kumuthu, is on site for asurprise inspection. He chooses the box of 30 and observes how many items are defective. What is the probabilitythat no more than 2 of the 30 products do not work? Hint: a tree will be helpful.

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