1. Use the combinatorial formula to calculate the following (12): a) You have 6 coloured flags. You want to select 2 of them. How many ways can you select two flags? Use your calculator to calculate the following: b) How many unique hands of Crazy Eight's (8 cards dealt to each player) can be dealt from a standard 52 card deck? c) Which method do you prefer? Why? 2. Given the following scenarios, determine if they can be classed as binomial experiments. Explain your response (8). a) A trainer is teaching a young dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, the probability that the dolphin performs another trick is 10%, and the probability that the dolphin does not successfully perform any trick is 55%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. b) Seventy-four percent of individuals fail the bar exam on the first try. A group of 50 recent exam-takers are randomly selected, to see how many individuals passed. 3. During the regular local soccer league season, the main forward Aadya had the highest field goal rate in the league. She scored with 51.4% of her shots. Suppose you choose a random sample of 70 shots made by Aadya during the this season. Let T = the number of shots that scored points. (10) a) What is the probability distribution for T? Explain why you believe it to be so. (4) b) Using the formulas, calculate the: (i) mean and (ii) standard deviation of T. (6) 4. A singer at Shinedown Classical has learned from long experience that the probability that a mic accident will occur while he is singing a song is 0.40. During one hour, the singer sings 5 songs. Make a graph of the binomial probability distribution that shows the probabilities that 0, 1, 2, 3, 4, or all 5 songs experiencing mic outages. (10) Hint: q = 0.40 ... p = ? ... n = ? 5. An ice-skating rink specifies that the average (mean) amount of ice on the rink should 6cm thick, with a standard deviation of only 0.5cm. An inspector picks out a random spot on the ice, and measures 4.1cm of ice. Assume that the amount of ice on the rink follows a normal distribution. If the ice on the rink is below the mean by more than 2 standard deviations, the ice-skating rink is in danger of losing its license. Is the rink safe to remain open? (10) 6. Crazy Keyboardz Inc., sells an expensive keyboard package, including a pre-built custom keyboard. Because the keyboard is so expensive, the company wants to advertise an impressive guarantee for the sound level (the keyboard promises to be very quiet) of the keyboard for 20 months. The guarantee policy will refund the full purchase price if the keyboard fails (becomes noisy) during the guarantee period. The market research department has done tests that show that the mean life for the keyboard is 20 months, with standard deviation of 3 months. After 20 months, most keyboards become loud and can break. The keyboard life is normally distributed. How long can the guarantee period be if management does not want to refund the purchase price on more than 5% of the Crazy Keyboardz packages? (10) Hint: Use the Normal Distribution Table.