Get on the boat! A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable Y = money collected (in dollars) on a randomly selected ferry trip is shown here.

Part (a). Find$P{(Y<20)}^{P(Y<20)}$ . Interpret this result.

Part (b). Express the event “at least $20 is collected” in terms of Y. What is the probability of this event?

If two events cannot occur at the same time, they are disjoint or mutually exclusive.

Addition rule for events that are disjoint or mutually exclusive:

$P\left(AUB\right)=P\left(A\right)+P\left(B\right)$

The following probabilities are provided by the table above.

localid="1653999013849" $$\begin{gathered} P(Y=0)=0.03 \\ P(Y=5)=0.05 \\ P(Y=10)=0.08 \\ P(Y=15)=0.16 \end{gathered}$$

Because collecting two different amounts for the same ferry trip is not possible. The gatherings are mutually exclusive.

For mutually exclusive events, apply the addition rule:

$$\begin{gathered} P(Y<20)=P(Y=0)+P(Y=5)+P(Y=10)+P(Y=15) \\ P(Y<20)=0.02+0.05+0.08+0.16 \\ P(Y<20)=0.31 \\ P(Y<20)=31\% \end{gathered}$$

This implies that about 31% of all ferry trips result in less than 20 dollars being collected.

If two events cannot occur at the same time, they are disjoint or mutually exclusive.

Addition rule for events that are disjoint or mutually exclusive:

$P\left(AUB\right)=P\left(A\right)+P\left(B\right)$

Because Y represents the money collected from a random ferry trip.$Y\ge 20$represents "at least $20 is collected."

The table above shows the following probabilities:

$$\begin{gathered} P(Y=20)0.27 \\ P(Y=25)=0.42 \end{gathered}$$

The events are mutually exclusive because it is not possible to collect two different amounts for the same ferry trip.

For mutually exclusive events, apply the addition rule:

$$\begin{gathered} P(Y\ge 20)=P(Y20)+P(Y-25) \\ P(Y\ge 20)=0.27+0.42 \\ P(Y\ge 20)=0.69 \\ P(Y\ge 20)=69\% \end{gathered}$$