Question

Skee Ball Ana is a dedicated Skee Ball player (see photo) who always rolls for the 50- point slot. The probability distribution of Ana’s score X on a randomly selected roll of the ball is shown here.

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

Part (b). Express the event “Anna scores at most 20” in terms of X. What is the probability of this event?

Short answer

Part (a) 41%

Part (b) 59%

Step-by-step solution

Part (a) Step 1. Given information.

Score	10	20	30	40	50
Probability	0.32	0.27	0.19	0.15	0.07

Part (a) Step 2.   P(Y<20) P(Y<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:

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

Score	10	20	30	40	50
Probability	0.32	0.27	0.19	0.15	0.07

The following probabilities are provided by the table above.

Because two different scores for the same game are not possible, the two events are mutually exclusive.

For mutually exclusive events, apply the addition rule:

$$\begin{gathered} P(X>20)=P(X=30)+P(X=40)+P(X=50) \\ P(X>20)=0.19+0.15+0.07 \\ P(X>20)=0.41 \\ P(X>20)=41\% \end{gathered}$$

A score of more than 20 is obtained in approximately 41% of the games.

Part (b) Step 1. Express the event “Anna scores at most 20” in terms of X. 

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:

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

Score	10	20	30	40	50
Probability	0.32	0.27	0.19	0.15	0.07

Because X represents the score on a randomly chosen roll of a ball, a score of at most 20 can be represented by$x\le 20$

The table above shows the following probabilities:

$$\begin{gathered} P(X=10)=0.32 \\ P(X=20)=0.27 \end{gathered}$$

Because two different scores for the same game are not possible, the two events are mutually exclusive.

For mutually exclusive events, apply the addition rule:

$$\begin{gathered} P(X\le 20)=P(X=10)+P(X-20) \\ P(X\le 20)=0.32+0.27 \\ P(X\le 20)=0.59 \\ P(X\le 20)=59\% \end{gathered}$$