Independent Events Again assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331, as in Exercise 1. If we want to find the probability of 20 females when 50 different speaking characters are randomly selected from a population of 1500 speaking characters, are the 50 selections independent? Using the 5% guideline for cumbersome calculations, can they be treated as being independent?

Verified step by step guidance

1. Step 1: Recall the definition of independent events. Two events are independent if the occurrence of one does not affect the probability of the other. In this case, we are determining whether selecting one speaking character affects the probability of selecting another.
2. Step 2: Use the 5% guideline for cumbersome calculations. This guideline states that if the sample size (n) is less than or equal to 5% of the population size (N), the selections can be treated as independent, even if they are technically dependent.
3. Step 3: Calculate 5% of the population size. The population size is 1500, so compute 0.05 × 1500 to determine the threshold for independence.
4. Step 4: Compare the sample size to the 5% threshold. The sample size is 50. If 50 is less than or equal to the value calculated in Step 3, the selections can be treated as independent.
5. Step 5: Conclude whether the selections can be treated as independent based on the comparison in Step 4. If the sample size is within the 5% threshold, the selections are treated as independent for the purposes of probability calculations.

Key Concepts

Independent Events

Independent events are those whose outcomes do not affect each other. In probability, two events A and B are independent if the occurrence of A does not change the probability of B occurring. For example, if selecting one character does not influence the selection of another, these selections can be considered independent.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of selecting a female character is given as 0.331. When calculating the probability of multiple independent events, the probabilities can be multiplied together to find the overall likelihood of a specific outcome.

Sampling Without Replacement

Sampling without replacement occurs when an item is selected from a population and not returned before the next selection. This can affect the independence of events, as the probabilities change with each selection. In this scenario, if the population of speaking characters is significantly larger than the sample size, the selections can often be treated as independent for practical calculations.