(i) According to research, 30% of offspring peas have green pods. Suppose a researcher chooses a random sample of 150 offspring peas. Let X = the number of offspring peas that have green pods.

(a) What is the probability distribution for X? Explain your answer in two sentences. (1 mark)

(b) Calculate the mean (to the nearest whole number) and standard deviation of X (to 4 d.p.). (1 mark)

(c) Use EXCEL to find the probability (to 4 d.p.) that the researcher has found exactly 40 offspring peas that have green pods. (1 mark)

(d) Use EXCEL to find the probability (to 4 d.p.) that the researcher has found more than 40 offspring peas that have green pods. (1 mark)

(ii) A total number of 194 tornadoes hit a city in one year. Let X= total number of tornadoes that hit the city in one year.

(a) What probability distribution would you use to model X? Explain your answer in two sentences. (1 mark)

(b) What is the probability (to 4 d.p.) that at most 80 tornadoes hit the city within half a year? (1 mark)

(iii) Human brain volumes (cm³) are known to be normally distributed with a mean of 1051 and a standard deviation of 5.4. Let V = brain volume of a human being in cm³.

(a) If one person is randomly chosen, find the probability (to 4 d.p.) that the brain volume is at least 1067. (1 mark)

(b) Between what two volumes (to nearest whole number) should 95% of all brain volumes be located? (1 mark)

(iv) The Ferries at Lyttleton Harbour are expected to arrive from Diamond Harbour every 30 minutes. Let X be the random variable for the number of minutes that somebody arriving at the wharf at a random time will have to wait before a ferry arrives.

(a) What probability distribution would you use to model X? Explain your choice in one or two sentences. (1 mark)

(b) Calculate the expected value of and the standard deviation of this distribution (to 3 d.p.). (1 mark)