A bag contains one each of red, blue, green, yellow and white marbles. Give the sample space for the following experiments.
A) one marble is drawn at random
B) one marble is drawn, but not replaced,and then a second marble is drawn.
Sample space (Ω) is the set of possible outcomes.
(a) the set of possible outcomes is the set each of the coloured marbles:
Ω={R,B,G,Y,W}
(b) Without replacement means that the two marbles are of different colours, so the sample space is
{RB,RG,RY,RW,BG,BY,BW,GY,GW,...}
There are 20 of them, order counts, for example, RG and GR are different outcomes.
A) Sample space for drawing one marble at random: {red, blue, green, yellow, white}
B) Sample space for drawing two marbles without replacement: {(red, blue), (red, green), (red, yellow), (red, white), (blue, red), (blue, green), (blue, yellow), (blue, white), (green, red), (green, blue), (green, yellow), (green, white), (yellow, red), (yellow, blue), (yellow, green), (yellow, white), (white, red), (white, blue), (white, green), (white, yellow)}
A) The sample space for drawing one marble at random consists of five possible outcomes: {red, blue, green, yellow, white}.
B) The sample space for drawing one marble without replacement and then a second marble is drawn consists of 20 possible outcomes. Since there are five marbles and the first marble drawn is not replaced before drawing the second marble, the number of outcomes is equal to the number of ways to choose two marbles out of five. This can be calculated using combinations or the binomial coefficient.
The sample space for this experiment is given by:
{(red, blue), (red, green), (red, yellow), (red, white),
(blue, green), (blue, yellow), (blue, white),
(green, yellow), (green, white),
(yellow, white)}.
To find the sample space for the given experiments, we need to consider all the possible outcomes that can occur.
A) Drawing one marble at random:
In this experiment, we have five different marbles in the bag. When one marble is drawn randomly, there are five possible outcomes because any one of the five marbles could be chosen. Therefore, the sample space for experiment A is {red, blue, green, yellow, white}.
B) Drawing one marble without replacement and then a second marble:
In this experiment, we need to consider that the first marble drawn is not replaced before drawing the second marble. Initially, there are five marbles in the bag.
When the first marble is drawn, there are five possible outcomes, as explained in experiment A. However, after the first marble is removed, there are only four marbles remaining in the bag for the second draw.
Therefore, for the second draw, there are only four possible outcomes.
Since the first and second draws are performed consecutively, we can multiply the number of outcomes for each draw. Hence, the sample space for experiment B is {red, blue, green, yellow, white} x {red, blue, green, yellow}, which results in the following sample space:
{red-red, red-blue, red-green, red-yellow, blue-red, blue-blue, blue-green, blue-yellow, green-red, green-blue, green-green, green-yellow, yellow-red, yellow-blue, yellow-green, yellow-yellow, white-red, white-blue, white-green, white-yellow}.
Note: The notation in the sample space indicates the color of the first marble drawn, followed by the color of the second marble drawn.