The next generation of miniaturized wireless capsules with active locomotion will require two miniature electric motors to maneuver each capsule. Suppose 10 motors have been fabricated but that, in spite of tests performed on the individual motors, 2 will not operate satisfactorily when placed into a capsule. To fabricate a new capsule, 2 motors will be randomly selected (that is, each pair of motors has the same chance of being selected). Find the probability that: a. Both motors will operate satisfactorily in the capsule. b. One motor will operate satisfactorily and the other will not.

Suppose 10 motors have been fabricated but that, in spite of tests performed on the individual motors, 2 will not operate satisfactorily when placed into a capsule. Then

there are 8 motors which operate satisfactorily when placed into a capsule;
there are 2 motors which do not operate satisfactorily when placed into a capsule.

To fabricate a new capsule, 2 motors will be randomly selected (that is, each pair of motors has the same chance of being selected).

A. The probability that both motors will operate satisfactorily in the capsule is

[tex]P=\dfrac{C^8_2}{C^{10}_2}=\dfrac{\frac{8!}{2!(8-2)!}}{\frac{10!}{2!(10-2)!}}=\dfrac{8!}{6!}\cdot \dfrac{8!}{10!}=7\cdot 8\cdot \dfrac{1}{9\cdot 10}=\dfrac{56}{90}=\dfrac{28}{45}[/tex]

B. The probability that one motor will operate satisfactorily and the other will not is

[tex]P=\dfrac{C^8_1\cdot C^2_1}{c^{10}_2}=\dfrac{\frac{8!}{1!(8-1)!}\cdot\frac{2!}{1!\cdot (2-1)!}}{\frac{10!}{2!(10-2)!}}=8\cdot 2\cdot \dfrac{2!\cdot 8!}{10!}=16\cdot \dfrac{2}{9\cdot 10}=\dfrac{32}{90}=\dfrac{16}{45}[/tex]