Unlocking Event Probability: Sample Space, Favorable Outcomes, and Experimental Probability

a Despite having only two colors, all five marbles are said to be distinguishable. That is why starting by labeling each of the marbles is a good option.

Since both marbles are randomly drawn, each outcome of the event will consist of two labels. For the first marble, there are $5$ possible outcomes. For the second marble, there are $4$ possible outcomes because one marble has already been removed from the bag.

Since the Diego draws a marble first and Emily draws a marble after him, the order in which the marbles are drawn matters. Therefore, the outcomes

{W_{1}, W_{2}}

and

{W_{2}, W_{1}}

are different. The following table lists all the possible outcomes for the event of drawing two marbles from the bag.

{W_{1}, W_{2}}, {W_{1}, B_{1}}, {W_{1}, B_{2}}, {W_{1}, B_{3}} {W_{2}, W_{1}}, {W_{2}, B_{1}}, {W_{2}, B_{2}}, {W_{2}, B_{3}} {B_{1}, W_{1}}, {B_{1}, W_{2}}, {B_{1}, B_{2}}, {B_{1}, B_{3}} {B_{2}, W_{1}}, {B_{2}, W_{2}}, {B_{2}, B_{1}}, {B_{2}, B_{3}} {B_{3}, W_{1}}, {B_{3}, W_{2}}, {B_{3}, B_{1}}, {B_{3}, B_{2}}

Consequently, there are a total of

20

possible outcomes in the sample space.

b The case in which two white marbles are drawn is represented by the outcomes containing only

W_{1}

and

W_{2} .

From Part A, it can be seen that there are only two such outcomes —

{W_{1}, W_{2}}

and

{W_{2}, W_{1}} .

c The case in which two black marbles are drawn is represented by the outcomes containing neither

W_{1}

nor

W_{2} .

Analyzing the sample space from Part A, six such outcomes can be found.

{B_{1}, B_{2}}, {B_{1}, B_{3}}, {B_{2}, B_{1}}, {B_{2}, B_{3}}, {B_{3}, B_{1}}, {B_{3}, B_{2}},

d The case in which one white and one black marble are drawn is represented by the outcomes containing one

W -

label and one

B -

label. From Part A, there are twelve such outcomes.

{W_{1}, B_{1}}, {W_{1}, B_{2}}, {W_{1}, B_{3}}, {W_{2}, B_{1}}, {W_{2}, B_{2}}, {W_{2}, B_{3}}, {B_{1}, W_{1}}, {B_{1}, W_{2}}, {B_{2}, W_{1}}, {B_{2}, W_{2}}, {B_{3}, W_{1}}, {B_{3}, W_{2}},

e There are an infinite number events that can be considered. For example, Diego and Emily could ask Paulina to give them two marbles each. In that case, the event would be the drawing of four marbles.

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Catch-Up and Review

Explore

Possible Outcomes of Target Shooting

Challenge

Probability of Winning a Prize

Discussion

Defining Experiments in Probability

Concept

Event

Example

Sample Space: Drawing Marbles From a Bag

Answer

Hint

Solution

Discussion

Probability of the Complement of an Event

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