Question

A bag contains four ​batteries, all of which are the same size and are equally likely to be selected. Each battery is a different brand. If you select three batteries at​ random, use the counting principle to determine how many points will be in the sample space if the batteries are selected.A) with replacement

b) without replacement

Answer 1

When selecting three batteries from a bag of four with replacement, there are 64 points in the sample space. Without replacement, the sample space contains 24 points, as the number of selections reduces the available options each time.

Step-by-step explanation:

When selecting three batteries from a bag of four, each being a different brand, the number of points in the sample space depends on whether the selection is with replacement or without replacement.

a. Sampling with replacement

For sampling with replacement, each time you select a battery, all four are available to choose from, since the selected battery is put back before the next selection. Therefore, for each of the three draws, there are 4 possibilities:

First draw: 4 possibilities

Second draw: 4 possibilities

Third draw: 4 possibilities

Using the counting principle, we multiply the number of possibilities for each draw: 4 × 4 × 4 = 64. So, there are 64 points in the sample space.

b. Sampling without replacement

For sampling without replacement, each selection reduces the number of available batteries:

First draw: 4 possibilities

Second draw: 3 possibilities (one battery has been removed)

Third draw: 2 possibilities (two batteries have been removed)

Again, applying the counting principle, we multiply the possibilities together: 4 × 3 × 2 = 24. Therefore, there are 24 points in the sample space when selecting without replacement.

Answer 2

a) 64

b) 24

Step-by-step explanation:

If the number of ways to do a thing = m

and the number of ways to do another thing = n

Total number of ways of doing the two things = m * n

A ) with replacement

So far the batteries are selected with replacement. After every selection, the number of batteries becomes complete agai.

Number of ways of selecting the first battery = 4

Number of ways of selecting the second battery = 4

Number of ways of selecting the third battery = 4

Total number of ways of selecting the three batteries = 4*4*4 = 64 ways

There will be 64 points in the sample space

b) without replacement

Since the batteries are replaced after every selection:

Number of ways of selecting first battery = 4

Number of ways of selecting second battery = 3

Number of ways of selecting third battery = 2

Total number of ways of selecting the three batteries = 4*3*2=24 ways

There will be 24 points in the sample space