Question

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 satisfactory when placed in 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 satisfactory in the capsule. B:One motor will operate satisfactory in the capsule and one will not.

Answer 1

a) There is a 61.6% probability that both motors will operate satisfactory in the capsule.

b) There is a 35.52% probability that one motor will operate satisfactory in the capsule and one will not.

Explanation:

We have 10 motors, of which

8 will operate correctly

2 will not operate correctly.

a)Find the probability that both motors will operate satisfactory in the capsule.

There are 10 motors, of which 8 will operate correctly. So the probability that the first motor will operate correctly is 80%.

There is no replacement, so now there are 9 motors, of which 7 operate correctly. So there is a
`(7)/(9) = 0.77` = 77% probability that the second motor will operate correctly.

So, the proability that both motors will operate satisfactory is:

`P = 0.8*0.77 = 0.616`

There is a 61.6% probability that both motors will operate satisfactory in the capsule.

b) One motor will operate satisfactory in the capsule and one will not.

There are two cases that satisfy this.

The first case is when the first motor operates satisfactory and the second does not.

There is a 80% probability that the first motor operates correctly. Then, there will be 9 motors, of which 2 will not operate correctly, meaning that the probability that the second capsule does not operate correctly is
`(2)/(9) = 0.222`. So, the probability that the first capsule operates correctly and the second does not is:

`P_(1) = 0.8*0.222 = 0.1776`

We can also have the first motor not operating satisfactory and the second operating.

2 out of 10 motors will not operate correctly, so the probability that the first motor does not operate correctly is 20%. Now, there are 9 motors, 8 of which will operate correctly. So, the probability that the second one will operate correctly in this case is
`(8)/(9) = 0.888`. This means that the probability that the first one does not operate satisfactory and the second does is

`P_(2) = 0.2*0.888 = 0.1776`

The probability that one motor will operate satisfactory in the capsule and one will not is

`P = P_(1) + P_(2) = 0.1776 + 0.1776 = 0.3552`