A bag of marbles has equal numbers of purple (P), blue (B), yellow (Y), and red (R) marbles. A marble is picked from the bag and replaced, then a second marble is picked. Complete parts a and b. a. Represent the possible outcomes in the sample space. Determine the theoretical probability the marbles picked will not include purple. (Type an integer or a fraction.) b. If the experiment is repeated 550 times, how many repetitions can you expect to result in no purple? repetitions (Round to the nearest whole number as needed.)

To solve your question, let’s break it down step by step.

Part a: Sample Space

Since there are 4 colors of marbles (Purple (P), Blue (B), Yellow (Y), Red (R)), and each marble is picked and replaced, the possible outcomes for picking the marbles can be represented as pairs. The sample space would include all combinations of two marble picks, which can be:

– (P, P), (P, B), (P, Y), (P, R)

– (B, P), (B, B), (B, Y), (B, R)

– (Y, P), (Y, B), (Y, Y), (Y, R)

– (R, P), (R, B), (R, Y), (R, R)

Therefore, the sample space consists of 16 outcomes as written above.

Probability of Not Including Purple

There are 3 colors that are not purple (B, Y, R). When picking two marbles, the combinations that do not include purple would be:

– (B, B), (B, Y), (B, R)

– (Y, B), (Y, Y), (Y, R)

– (R, B), (R, Y), (R, R)

So that’s a total of 9 outcomes without purple out of 16 total outcomes.

The theoretical probability, therefore, is:

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