91Ó°ÊÓ

A box contains 100 balls, of which rare red. Suppose that the balls are drawn from the box one at a time, at random,without replacement. Determine (a) the probability that the first ball drawn will be red; (b) the probability that the 50th ball drawn will be red, and (c) the probability that the last ball drawn will be red.

Question:Suppose that a point (X, Y ) is chosen at random from the circle S defined as follows:\({\rm{S}} = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right\}\)

a. Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y .

b. Are X and Y independent

Three six-sided dice are rolled. The six sides of each die are numbered\(1 - 6\). Let A be the event that the first die shows an even number, let B be the event that the second die shows an even number, and let C be the event that the third die shows an even number. Also, for each\(i = 1,2,...,6\), let\({A_i}\)be the event that the first die shows the number i, let \({B_i}\) be the event that the second die shows the number i, and let \({C_i}\)be the event that the third die shows the number i. Express each of the following events in terms of the named events described above:

a. The event that all three dice show even numbers

b. The event that no die shows an even number

c. The event that at least one die shows an odd number

d. The event that at most two dice show odd numbers

e. The event that the sum of the three dices is no greater than 5.

Suppose that two boys named Davis, three boys named Jones, and four boys named Smith are seated at random in a row containing nine seats. What is the probability that the Davis boys will occupy the first two seats in the row, the Jones boys will occupy the next three seats, and the Smith boys will occupy the last four seats?

A box contains 24 light bulbs, of which two are defective. If a person selects 10 bulbs at random, without replacement, what is the probability that both defective bulbs will be selected?

For two arbitrary events A and B, prove that

\({\bf{Pr}}\left( {\bf{A}} \right){\rm{ }} = {\rm{ }}{\bf{Pr}}({\bf{A}} \cap {\bf{B}}){\rm{ }} + {\rm{ }}{\bf{Pr}}({\bf{A}} \cap {{\bf{B}}^c}).\)

Letnandkbe positive integers such that bothnandn−kare large. Use Stirling’s formula to write as simple an approximation as you can forP_n,k.

A point (x,y) is to be selected from the squareScontaining all points (x,y) such that 0≤x≤1 and 0≤y≤1. Suppose that the probability that the selected point willbelong to each specified subset ofSis equal to the area ofthat subset. Find the probability of each of the followingsubsets:

(a) the subset of points such that\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y - \frac{1}{2}} \right)^2} \ge \frac{1}{4}\)

(b) the subset of points such that\(\frac{1}{2} < x + y < \frac{3}{2}\)

(c) the subset of points such that\(y \le 1 - {x^2}\)

(d) the subsetof points such thatx=y.

Suppose that 10 cards, of which five are red and five are green, are placed at random in 10 envelopes, of which five are red and five are green. Determine the probability that exactly x envelopes will contain a card with a matching colour (x = 0, 1, ... , 10).

Suppose that 35 people are divided in a random manner into two teams in such a way that one team contains10 people and the other team contains 25 people. What is the probability that two particular people A and B will be on the same team?