\begin{tabular}{l} Casey is going to wear a gray sportcoat and is trying to decide what tie he should wear to work. In his closet, he has \ 36 ties, 14 of which he feels go well with the sportcoat. If Casey selects one tie at random, determine the probablity \ and the odds of the tie going well or not going well with the sportcoat. \ (Simplify your answer. Type an integer or a fraction.) \ The probability the tie will not go well with the jacket is $$ \frac{7}{18} $$. \ (Simplify your answer. Type an integer or a fraction.) \ The odds against the tie going well with the jacket is $$ 11: 7 $$. \ (Simplify your answer. Type a ratio using a colon.) \ The odds in favor of the tie going well with the jacket is \ (Simplify your answer. Type a ratio using a colon.) \ (the goes well with the jacket is \ \hline\end{tabular}

Question

UpStudy AI · Accepted Answer

To solve the problem, we need to determine the probability and odds related to Casey's tie selection.

### Step 1: Calculate the Probability of the Tie Going Well with the Sportcoat

Casey has a total of 36 ties, and 14 of them go well with the sportcoat. The probability $$ P(G) $$ that a randomly selected tie goes well with the sportcoat is given by the formula:

$$
P(G) = \frac{\text{Number of ties that go well}}{\text{Total number of ties}} = \frac{14}{36}
$$

Now, we simplify $$ \frac{14}{36} $$:

$$
\frac{14}{36} = \frac{7}{18}
$$

### Step 2: Calculate the Probability of the Tie Not Going Well with the Sportcoat

The probability $$ P(NG) $$ that a randomly selected tie does not go well with the sportcoat is:

$$
P(NG) = 1 - P(G) = 1 - \frac{7}{18} = \frac{18 - 7}{18} = \frac{11}{18}
$$

### Step 3: Calculate the Odds Against the Tie Going Well with the Sportcoat

The odds against an event is calculated as the ratio of the probability that the event does not occur to the probability that it does occur. Thus, the odds against the tie going well is:

$$
\text{Odds against} = \frac{P(NG)}{P(G)} = \frac{\frac{11}{18}}{\frac{7}{18}} = \frac{11}{7}
$$

This means the odds against the tie going well with the jacket is $$ 11:7 $$.

### Step 4: Calculate the Odds in Favor of the Tie Going Well with the Sportcoat

The odds in favor of an event is the ratio of the probability that the event occurs to the probability that it does not occur. Thus, the odds in favor of the tie going well is:

$$
\text{Odds in favor} = \frac{P(G)}{P(NG)} = \frac{\frac{7}{18}}{\frac{11}{18}} = \frac{7}{11}
$$

### Final Answers

1. The probability the tie will go well with the jacket is $$ \frac{7}{18} $$.
2. The probability the tie will not go well with the jacket is $$ \frac{11}{18} $$.
3. The odds against the tie going well with the jacket is $$ 11:7 $$.
4. The odds in favor of the tie going well with the jacket is $$ 7:11 $$.
The probability the tie will go well with the jacket is $$ \frac{7}{18} $$. The probability the tie will not go well with the jacket is $$ \frac{11}{18} $$. The odds against the tie going well with the jacket are $$ 11:7 $$. The odds in favor of the tie going well with the jacket are $$ 7:11 $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Probability Questions