From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls.

A box contains 2 red, 3 white, and 4 green balls. Two balls are drawn out of the box in succession without replacement. What is the probability that both balls are the same color?

A box has b black balls and r red balls. Balls are drawn from the box one at a time without replacement. Find the probability that the first black ball selected is drawn at the nth trial.

A continuous random variable X that can assume values between x=2 and x=5 has a density function given by f(x)=2(1+x)/27. Find (a) P(X$\lt$4), (b) P(3 $\leq X \lt$ 4).

Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modeling these errors. Suppose the measurement error X of a certain physical quantity is decided by the density function

$$f(x)=k(3-x^2), \text{ for } -1 \leq x \leq 1,$$

$$f(x)=0, \text{ elsewhere.}$$

(a) Determine k that renders f(x) a valid density function. (b) Find the probability that a random error in measurement is less than 1/2. (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., |x|) exceeds 0.8. What is the probability that this occurs?

Three cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades.