(d) Justify why a is greater than % without doing any computation. {We are looking for a short answer giving a heuristic sense of why this is so.) (e) Use R simulation to estimate the smallest n. for which P(B) r .9. For these simulations, let the number of triaLs be 10000. For this value of n, repeat the simulation a few times to verify that it always gives similar results. Using 10000 trials you saw very little variation in the estimate of P(B). Try this again using 30 trials and verify that the estimated probabilities are much more variable. (f) Find an exact formula for P(B}. (g) Use R simulation to estimate the smallest n for which 13(0) :r- .5. Again use 10000 trials. You will nd that two values of n are equally plausible values. You may pick either one for your answer. Note that it is much harder to nd an exact formula for P{C}, so simulation is especially handy. (h) In real life, some birthdays are more common than others. Click on the first graph here to see the data