Question

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?

Short answer

Part (a) 31%

Part (b) 69%

Step-by-step solution

Part (a) Step 1. Given information.

Money collected	0	5	10	15	20	25
Probability$\left(Y=y\right)$	0.02	0.05	0.08	0.16	0.27	0.42

Part (a) Step 2. P(X>20)P(X>20). Interpret this result 

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:

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

Money collected	0	5	10	15	20	25
Probability $P\left(Y=y\right)$	0.02	0.05	0.08	0.16	0.27	0.42

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.

Part (b) Step 1. The event “at least $20 is collected” in terms of Y 

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:

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

Money collected	0	5	10	15	20	25
Probability $\left(Y=y\right)$	0.02	0.05	0.08	0.16	0.27	0.42

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:

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