Lesson 6: Find Probabilities of Compound Events

Property

The probability of a compound event is the ratio of the number of favorable outcomes to the total number of possible outcomes.

Examples

• A coin is flipped and a standard six-sided die is rolled. The probability of getting tails and a number greater than 4 is $\frac{2}{12} = \frac{1}{6}$. There are 2 favorable outcomes (Tails, 5; Tails, 6) out of 12 possible outcomes.
• A spinner has 3 equal sections (Red, Blue, Green) and another spinner has 2 equal sections (1, 2). The probability of landing on Blue and 1 is $\frac{1}{6}$. There is 1 favorable outcome (Blue, 1) out of $3 \times 2 = 6$ possible outcomes.

Explanation

Calculating the probability of a compound event follows the same principle as for a simple event. First, determine the total number of possible outcomes, often by using the Fundamental Counting Principle, a tree diagram, or a table. Next, count the number of outcomes that are considered favorable for the event. The probability is the fraction formed by placing the number of favorable outcomes in the numerator and the total number of outcomes in the denominator.

Property

The probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes. A tree diagram helps visualize all possible outcomes of a compound event.

Examples

• A coin is tossed twice. The tree diagram shows 4 possible outcomes: HH, HT, TH, TT. The probability of getting exactly one tail is $\frac{2}{4}$ or $\frac{1}{2}$, as there are two favorable outcomes (HT, TH).
• A number cube is rolled and a coin is tossed. The tree diagram shows 12 possible outcomes. The probability of rolling a number less than 3 and tossing heads is $\frac{2}{12}$ or $\frac{1}{6}$, as there are two favorable outcomes (1H, 2H).

Explanation

A tree diagram is a tool used to map out the sample space of a compound event. Each unique path from the start to an endpoint of the diagram represents one possible outcome. To find the probability of a specific event, count the number of paths that meet the event''s criteria (favorable outcomes) and divide by the total number of paths (total outcomes).

Property

To find the probability of a compound event using an organized list, first list all possible outcomes in the sample space. Then, count the number of favorable outcomes and the total number of outcomes. The probability is the ratio of these two counts.

Examples

• Suppose you toss a coin and roll a six-sided die. The organized list of outcomes is: (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6). The probability of getting tails and an even number is $P(\text{Tails and Even}) = \frac{3}{12} = \frac{1}{4}$.
• A bag has one red (R) and one blue (B) marble. A second bag has one green (G), one yellow (Y), and one orange (O) marble. The organized list for picking one marble from each bag is: (R,G), (R,Y), (R,O), (B,G), (B,Y), (B,O). The probability of picking a blue marble and an orange marble is $P(\text{B and O}) = \frac{1}{6}$.

Explanation

An organized list helps you visualize the entire sample space for a compound event. By systematically writing down every possible combination of outcomes, you can ensure no possibilities are missed. Once the list is complete, you can directly count the total number of outcomes and the specific outcomes that match the event you are interested in. This method provides a clear and straightforward way to calculate the probability by forming a fraction with these counts.