Question

Draw an apprapriate tren diagram, and use the multiplication principle to calculate the probablitios of all the outcomes. HINT [SEe Example 3.] Your aute rental eampany rents out 35 smali cars, 24 luxury sedans, and 41 slightly damaged "budget" vehicles. The small can break down 13% of the time, the luarury sedans break donn BW of the time, and the "budget" cars break down 40% of the time. P(smali and breaks down )= P(smali and does not break down) = P(Luxury and breaks down) = P(Luxury and dons not break down )= P(budget and breaks down) = P[Ridget and dons not brnak donn) =

Answer 1

1. \(P(\text{Small and Breaks Down}) = 0.13 \times 0.87 = 0.1131\)

2. \(P(\text{Small and Does Not Break Down}) = 0.87 \times 0.87 = 0.7569\)

3. \(P(\text{Luxury and Breaks Down}) = 0.08 \times 0.92 = 0.0736\)

4. \(P(\text{Luxury and Does Not Break Down}) = 0.92 \times 0.92 = 0.8464\)

5. \(P(\text{Budget and Breaks Down}) = 0.40 \times 0.60 = 0.24\)

6. \(P(\text{Budget and Does Not Break Down}) = 0.60 \times 0.60 = 0.36\)

Step-by-step explanation:

The multiplication principle allows us to calculate the probabilities of independent events occurring together. For small cars, the probability of breaking down is 0.13, so the probability of not breaking down is \(1 - 0.13 = 0.87\). Therefore, the probability of a small car both breaking down and not breaking down is calculated by multiplying these individual probabilities, yielding \(0.13 \times 0.87 = 0.1131\) and \(0.87 \times 0.87 = 0.7569\), respectively.

Similarly, for luxury sedans, the probability of breaking down is 0.08, and the probability of not breaking down is \(1 - 0.08 = 0.92\). The probabilities of a luxury sedan breaking down and not breaking down are \(0.08 \times 0.92 = 0.0736\) and \(0.92 \times 0.92 = 0.8464\), respectively.

For budget vehicles, the probability of breaking down is 0.40, and the probability of not breaking down is \(1 - 0.40 = 0.60\). The probabilities of a budget vehicle breaking down and not breaking down are \(0.40 \times 0.60 = 0.24\) and \(0.60 \times 0.60 = 0.36\), respectively.

Answer 2

To determine car rental probabilities based on breakdown rates, utilize the multiplication principle in a tree diagram. Calculate the likelihood of each outcome by considering the number of vehicles and breakdown probabilities. Normalize results by dividing by the total number of vehicles (100).

Step-by-step explanation:

To compute the probabilities of car rental outcomes, use breakdown rates and vehicle counts.

Construct a tree diagram and employ the multiplication principle for probabilities.

For a small car, determine the likelihood of breakdown (P(Small ∧ Breaks Down)) as the product of small car quantity (35) and breakdown probability (0.13).

Calculate the probability of no breakdown (P(Small ∧ Does Not Break Down)) similarly.

Apply the same approach to luxury sedans and budget cars.

Normalize probabilities by dividing each by the total vehicle count (100).

For instance, P(Small ∧ Breaks Down) becomes (35 × 0.13) / 100. Extend this process to establish the complete probability distribution for car rental outcomes.