Exercise 4 Page 675

Knowing whether events are independent or dependent can affect how we approach analyzing the probability of an outcome. Let's consider two cases and how we can tell if the events are independent or dependent in each case.

1. We know the probabilities of the events.
2. We do not know the probabilities of the events.

Probabilities Are Known

Assume we have events $A$ and $B.$ If we are given the probabilities $P(A),$ $P(B),$ and $P(A\text{ and }B),$ we can determine if the events are independent by checking that the events satisfy the formula for the probability of independent events. If the events do not satisfy this probability, the events are dependent.

Probabilities Are Unknown

If we are not given the probabilities, we have to identify if the outcome of one of the events affects the outcome or the probability of the other event. We will illustrate this using the pieces of paper from Exploration $1b.$ In the beginning, we have $6$ choices for our first selection.

Imagine that we chose the fifth piece of paper. Since we are selecting without replacement, after we select a piece of paper, we cannot select that piece again.

Since we only have $5$ pieces of paper left, there are only $5$ from which to choose for our second choice. This changes the probability of choosing any of the remaining pieces.

Summary

To know if two consecutive events are dependent, we can either look at their probabilities or determine whether the outcome of the first event will affect the possible outcomes of the second event.

• If the possible outcomes of the second event are changed, the events are dependent.
• If the possible outcomes of the second event remain the same, the events are independent.