Question

Determine whether the two given events are independent. Two planets are selected, without replacement, from the list on the right. The events are "the first selected planet is closer than Neptune" and "the second selected planet is farther than Mars."

Mean Distance of Planets from the Sun:
Mercury 58
Venus 108
Earth 150
Mars 228
Jupiter 778
Saturn 1430
Uranus 2870
Neptune 4500​

Answer 1

The events "the first selected planet is closer than Neptune" and "the second selected planet is farther than Mars" are not independent because the first selection affects the remaining choices for the second event, altering its probability.

Step-by-step explanation:

To determine whether the two events are independent, we have to check if the occurrence of the first event affects the probability of the second one occurring. The events are "the first selected planet is closer than Neptune" and "the second selected planet is farther than Mars."

Let's identify the planets closer than Neptune: Mercury, Venus, Earth, Mars, Jupiter, Saturn, and Uranus. And the planets farther than Mars: Jupiter, Saturn, Uranus, and Neptune.

Initially, there are seven choices that satisfy the first event. Once one of these planets is selected, without replacement, there are fewer planets remaining which affects the probability of the second event. Therefore, the probability of selecting a planet farther than Mars as the second planet is affected by the first selection. Consequently, the two events are not independent.

As an example, if Jupiter is selected first, it cannot be selected again, changing the probability that the second planet will be farther than Mars (since one option is now gone). This interdependence shows that the selection results of the two planets are not independent.