Calculating Compound Events: Understanding Probability with Independent and Dependent Events

compound event

Compound events refer to situations in probability where multiple events occur together or sequentially

Compound events refer to situations in probability where multiple events occur together or sequentially. These events can be dependent or independent, and the probabilities associated with each event can be calculated using various methods.

To understand compound events, we need to first understand the concepts of independent and dependent events:

1. Independent Events: In independent events, the occurrence or outcome of one event does not affect the occurrence or outcome of the other events. For example, flipping a coin or rolling a dice are independent events. When calculating the probability of independent events occurring together, we multiply the probabilities of each individual event.

2. Dependent Events: In dependent events, the occurrence or outcome of one event affects the occurrence or outcome of the other events. For example, drawing cards from a deck without replacing them is a dependent event. When calculating the probability of dependent events occurring together, we multiply the probabilities for each event, taking into account the outcomes of the previous events.

To calculate the probability of a compound event, we need to follow these steps:

Step 1: Determine whether the events are independent or dependent.

Step 2: Calculate the probability of each individual event.

Step 3 (for independent events): Multiply the probabilities of each individual event to find the probability of the compound event.

Step 3 (for dependent events): Multiply the conditional probabilities of each event, considering the outcomes of previous events, to find the probability of the compound event.

Step 4: If desired, express the probability as a fraction, decimal, or percentage.

Let’s consider an example to illustrate these steps:

Example: A jar contains 6 red marbles and 4 blue marbles. What is the probability of selecting a red marble, replacing it, and then selecting a blue marble?

Solution:
Step 1: These events are independent since each time a marble is selected, it is replaced before the next selection.

Step 2: The probability of selecting a red marble on the first draw is 6/10, as there are 6 red marbles out of 10 total marbles.

Step 2 (continued): After replacing the red marble, the probability of selecting a blue marble on the second draw is 4/10, as there are 4 blue marbles remaining out of 10 total marbles.

Step 3 (for independent events): Multiply the probabilities of each individual event:
(6/10) * (4/10) = 24/100 = 0.24

Step 4: The probability of selecting a red marble, replacing it, and then selecting a blue marble is 0.24 or 24/100 or 24%.

So, in this example, the probability of the compound event occurring is 0.24 or 24%.

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