A bag of marbles has equal numbers of red (R), purple (P), blue (B), and orange (O) 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. Choose the correct answer below.
Click here to view sample space a.
Click here to view sample space c.
Click here to view sample space d.
Click here to view sample space b.
Determine the theoretical probability the marbles picked will not include purple.
(Type an integer or a fraction.)
To solve this problem, let’s break it down step-by-step.
a. The sample space represents all possible outcomes when picking two marbles. Since we have four colors: Red (R), Purple (P), Blue (B), and Orange (O), each marble picked can be any of these colors.
When we pick two marbles (with replacement), the sample space will include all the combinations of the two picks. The possible pairs (considering order) can be represented as follows:
– RR, RP, RB, RO
– PR, PP, PB, PO
– BR, BP, BB, BO
– OR, OP, OB, OO
So, the complete sample space consists of 16 outcomes:
{ RR, RP, RB, RO, PR, PP, PB, PO, BR, BP, BB, BO, OR, OP, OB, OO }
b. To determine the theoretical probability that the marbles picked will not include purple (P):
1. First, we identify the outcomes that do not include purple. The acceptable outcomes are:
– RR, RB, RO
– BR, BB, BO
– OR, OB, OO
This results in 9 outcomes that do not have purple.
2. The total number of outcomes is 16, as we identified above.
3. The probability is calculated as the number of favorable outcomes divided by the total number of outcomes:
[
P(text{not purple})