the question is: A shrub that is purchased at a local nursery has probability .80 of surviving the first year. If a customer pruchases 4 shrubs, what is the probability that 2 or more will survive the first year? If a landscaper purchases 200 shrubs, give a likely range for the number that will survive the first year. Help!

A few hints:

Since you want to find the probability that two or more shrubs survive, find the probability that no shrub survives+probability that exactly one shrub out of the four survives. Then subtract this number from 1.

For the second part, it's easy to see that the expected number of shrubs surviving the first year is 0.8*200=160 (expectation of a Bernoulli random variable, here the number of survivals). The s.d of number of survivals is [200*0.8*(1-0.8)]^0.5=5.66.

Now to find a *range* for the likely number of survivals, you need to know that the number of survivals becomes approximately normally distributed when the total number of shrubs is large.

So a >99% (to be precise 99.73%) confidence interval for the number of survivals would be (160-3*5.66, 160+3*5.66) where 3 is the standard normal variable corresponding to 99.73%. In other words, 99.73% of the values of a normal distribution lie within 3 standard deviations of the mean. In this case, there is a 99.73% chance that the number of survivals lies between 160-3*5.66 and 160+3*5.66. Hence this is a very "likely" range.