Use a sample space to determine whether the events are independent. There are three green apples and one red apple in a bowl. You randomly select one apple to eat now and another apple to eat with lunch. Determine whether randomly selecting a green apple first and randomly selecting a green apple second are independent events.
Question
Use a sample space to determine whether the events are independent. There are three green apples and one red apple in a bowl. You randomly select one apple to eat now and another apple to eat with lunch. Determine whether randomly selecting a green apple first and randomly selecting a green apple second are independent events.
Solution
To determine if the events are independent, we need to check if the probability of the second event (selecting a green apple second) changes depending on the outcome of the first event (selecting a green apple first).
Step 1: Identify the total number of outcomes. There are 4 apples in total, so there are 4 possible outcomes for the first selection and 3 possible outcomes for the second selection.
Step 2: Calculate the probability of each event separately. The probability of selecting a green apple first is 3/4 (3 green apples out of 4 total apples). After one apple is removed, the probability of selecting a green apple second is 2/3 (2 green apples out of 3 total apples left).
Step 3: Calculate the probability of both events happening together. If a green apple is selected first, there are 2 green apples and 3 total apples left. So, the probability of selecting a green apple first and second is (3/4) * (2/3) = 1/2.
Step 4: Check if the probability of both events happening together is equal to the product of the probabilities of each event happening separately. If they are equal, the events are independent. In this case, the probability of both events happening together (1/2) is not equal to the product of the probabilities of each event happening separately (3/4 * 2/3 = 1/2).
Therefore, the events are not independent. The outcome of the first event (selecting a green apple first) does affect the probability of the second event (selecting a green apple second).
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