Question

Select the situation that contain independent events.Options:The favorite fruit of five randomly selected shoppers in the produce section of a grocery store.Drawing an ace and then drawing a spade from a standard deck of 52 cards without replacing the first card.Smoking cigarettes and getting lung cancer.Training as a long distance runner and winning a marathon.

Answer 1

- Two events A and B are said to be independent if the occurrence of one does not affect the probability of occurrence of the other.

- With the above definition, we can proceed to eliminate the options that have dependent events.

Option 1:

- We have no prior knowledge of 5 randomly selected shoppers. Thus, the probability that shopper 1's favorite fruit is an Orange will in no way have an effect on the probability of shopper 2's favorite fruit being any other fruit or even an orange.

Therefore, this Option contains Independence

Option 2:

- When an ace is drawn, the number of cards reduces to 51 in total. This implies that if a spade is chosen next, the probability that this second event of pulling a spade occurred would be different from its initial probability because the total number of cards has been reduced by 1. That is, the probability of choosing every other card increased slightly because we initially took out the ace with no replacement. That is,

`\begin{gathered} P(ace)=(13)/(52) \\ (\text{ This is the probability of initially choosing the ace}) \\ \\ P(spade)=(13)/(52) \\ (\text{ This is the probability of choosing a spade when the deck of card is complete}) \\ \\ \\ \text{ The probability of choosing a spade when an ace has been removed becomes} \\ P(spade)=(13)/(51) \end{gathered}`

- Therefore, this Option contains dependence

Option 3:

- Whenever a person smokes, they are doing damage to their lungs. This can directly lead to lung cancer.

Thus, we can conclude that the likelihood of getting lung cancer increases when a person smokes. And if likelihood increases, probability increases. This implies that the event of smoking affects (i.e. it increases) the probability of getting lung cancer.

- Therefore, this Option also contains dependence.

Option 4:

- If a person trains to run long distances, he builds his body and endurance for long-distance sprints. If the person hadn't done the training, he would not be able to run long distances. This directly implies that the probability of a person winning a marathon increases when they train to run long distances because they are better equipped in body and mind to participate in such long-distance sprints.

- Therefore, this Option also contains dependence.

Final Answer

OPTION 1 is the only option that contains Independence