Probability - Multiplication and Addition rule

SOURCE:COMPETITION
Number of Problems: 14. : (Book)

A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?

$extbf{(A) } frac{1}{8} qquad extbf{(B) } frac{5}{32} qquad extbf{(C) } frac{9}{32} qquad extbf{(D) } frac{3}{8} ...$

One fair die has faces $1$ , $1$ , $2$ , $2$ , $3$ , $3$ and another has faces $4$ , $4$ , $5$ , $5$ , $6$ , $6$ . The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

$mathrm{(A)} frac{1}{3} qquad mathrm{(B)} frac{4}{9} qquad mathrm{(C)} frac{1}{2} qquad mathrm{(D)} frac{5}{9} qqu...$

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

$mathrm{(A)} left(frac{1}{12} ight)^{12} qquad mathrm{(B)} left(frac{1}{6} ight)^{12} qquad mathrm{(C)} 2left(fra...$

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ $20$ or more?

$mathrm{(A)} {{{frac{1}{4}}}} qquad mathrm{(B)} {{{frac{2}{7}}}} qquad mathrm{(C)} {{{frac{3}{7}}}} qquad mathrm{...$

10.

Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

$mathrm{(A) } frac{29}{128} qquad mathrm{(B) } frac{23}{128} qquad mathrm{(C) } frac14 qquad mathrm{(D) } f...$

11.

For a particular peculiar pair of dice, the probabilities of rolling $1$ , $2$ , $3$ , $4$ , $5$ , and $6$ , on each die are in the ratio $1:2:3:4:5:6$ . What is the probability of rolling a total of $7$ on the two dice?

$mathrm{(A) } frac{4}{63}qquad mathrm{(B) } frac{1}{8}qquad mathrm{(C) } frac{8}{63}qquad mathrm{(D) } frac...$

12.

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

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$extbf{(A) } frac{1}{3} qquad extbf{(B) } frac{4}{9} qquad extbf{(C) } frac{1}{2} qquad extbf{(D) } frac{5}{9} qq...$

13.

Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?

$mathrm{(A)} 1/12qquadmathrm{(B)} 1/10qquadmathrm{(C)} 1/6qquadmathrm{(D)} 1/3qquadmathrm{(E)} 1/2$

14.

Integers $a, b, c,$ and $d$ , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?

$mathrm{(A)} frac 38qquad mathrm{(B)} frac 7{16}qquad mathrm{(C)} frac 12qquad mathrm{(D)} frac 9{16}qquad ma...$