Four balls in an urn are identical except for color; one is...

Four balls in an urn are identical except for color; one is red $(R)$, one white $(W)$, one yellow $(Y)$, and the fourth is blue $(B)$. The experiment is to pick a ball from the urn and then, without replacing it, pick a second ball. Use a tree diagram to find $P$ (at least one $Y$ ball).

Key Concepts

Tree Diagram

A tree diagram is a visual tool used to map out all the possible outcomes of a sequential process or experiment. It systematically represents events as branches, helping in the organization of information and in the calculation of probabilities for sequences of events.

Sequential Experiments and Dependent Events

When dealing with experiments where outcomes occur in sequence, the outcome of one event can affect the outcomes of subsequent events, particularly when the process is carried out without replacement. This means that the events are dependent, and the probabilities must be adjusted at each step based on the previous outcomes.

Sample Space

The sample space is the complete set of all possible outcomes of an experiment. In sequential experiments, the sample space is structured and can be represented by all the branches in a tree diagram, which facilitates the enumeration and calculation of probabilities for various events.

Probability Calculation

Probability calculation involves assigning and summing the probabilities of individual events or paths that lead to a specific outcome within the sample space. When events are mutually exclusive, their probabilities can be summed to determine the likelihood of a particular compound event.

Complement Rule

The complement rule is a useful probability principle that states the probability of an event occurring is equal to one minus the probability of it not occurring. This rule is particularly helpful when it is easier to calculate the probability of the complementary event rather than the event in question.

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