There are 1000 pupils in a school. Find the probability that exactly 3 of them have their birthdays on 1 January, by using (a) B(1000, 1/365) (b) Po(1000/365) There are 5000 students in a university. Calculate the II probability that exactly 15 of them have their birthdays on 1 January by using (a) suitable binomial distribution approximation III (a) In a certain school, 30% of the students are in the age (b) suitable Poisson group 16-19. i Ten students are chosen at random. What is the probability that fewer than four of them are in the 16-19 age group? ii if the ten students were chosen by picking ten who were sitting together at lunch, explain why a binomial distribution might no longer have been suitable. (b) A factory makes large quantities of coloured sweets and it is known that on average 20% of the sweets are coloured green. A packet contains 20 sweets. Assuming that the packet forms a random sample of the sweets made by the factory, calculate the probability that exactly seven of the sweets are green. If you knew that, in fact, the sweets could have been green, red, orange or brown, would it have invalidated your calculation? IV (a) Find the mean of the random variables X and Y which have the following probability distribution (i) P(X = x) (ii) P(Y = y) (b) The random variable T has the probability distribution given in the following table 1 2 3 4 1/8 3/8 1/8 ¼ 1/8 X y -2 -1 1 2 3 0.15 0.25 0.3 0.05 0.2 0.05 1 3 4 6. 7 0.1 P (T= t) Find E (T) and Var (T) V Given that X - N (44, 25), find s, t, u and v (a) P (X< s) = 0.9808 (b) P(X > t) = 0.7704 (c) P(X > u) = 0.0495 (d) P(X< v) = 0.3336 0.2 0.1 0.2 0.1 0.2 0.1 VI X has normal distribution and P(X>73.05) = 0.0289. Given that the variance of the distribution is 18, find the mean.

Suppose 50% of the registered voters in a country are Republican if a sample of 598 voters is selected what is the probability that the simple proportion of republicans will be greater than 47% round your answer to 4 decimal places

The mean (or expected value) of a probability distribution can be calculated by first making the probability distribution. Then multiply the possible values of the Random Variable by their probabilities (X*P(X)) then add all the X*P(X) up.  On the TI 83/84 calculator you can  1) Enter the values of the RV's into List 1 (Press STAT, 1 (for Edit), highlist L1 and press CLEAR if needed, then enter values into the list) 2) Enter the probabilities into L2 (press > and then enter the values in the list. highlight L2 and press CLEAR first if needed). 3) 2nd MODE (to quit the list) 4) STAT, > (for CALC), select 1-Var Stats, List: L1 (2nd 1), FreqList: L2 (2nd 2).  Then calculate.  x¯ is the expected value (mean) of the frequency distribution.    A game is played. If you win you gain $4. If you lose you lose $10. The probability of winning is 0.747.  What is the expected value (mean)?

Suppose bird attacks follow a Poisson process and attacks occur at a rate of 4 attacks per week What is the probability of a bird free week (0 bird attacks)? In a period of 16 weeks, what is the probability of exactly 2 bird free weeks? What is the probability that it takes 16 weeks for there to be exactly 3 bird free weeks?