Question

In the state of Toyotia, the driving test consists of three parts. Candidates have to pass all three parts in order to get their licence. It is observed that pass rates for each part vary depending on whether the candidate is attempting the part for the first time or repeating it after a previous failure. Pass rates are given in the following table. Assume that a candidate's performance on any attempt at any test is independent of their performance on any other attempt. Syntax advice: Give your answers to the following questions as exact fractional expressions, not decimals. You do not have to simplify your fractions, for example, (1/2)+(34/56)+(789/10) is an allowable answer. Make sure you include all necessary brackets, for example, (1+2+3)/(4 ∗5+6).

(a) Find the probability that a candidate passes all three tests at the first attempt. Do not enter a decimal. Answer: (1) (b) Find the probability that a candidate passes all three tests in a total of exactly four attempts.

Answer 1

To find the probability of passing all three tests at the first attempt, multiply the pass rates. To find the probability of passing all three tests in four attempts, consider the combinations of passes and failures.

Step-by-step explanation:

To find the probability that a candidate passes all three tests at the first attempt, you need to multiply the pass rates for each part. Let's assume the pass rates for the three parts are P1, P2, and P3, respectively. The overall probability can be calculated as P = P1 * P2 * P3. Since the question doesn't provide the specific pass rates, I can't give you the exact value, but it should be a product of three fractions.

To find the probability that a candidate passes all three tests in a total of exactly four attempts, we need to account for the possible combinations of test parts passed and failed in each attempt. We can use the binomial coefficient formula to calculate the number of ways to arrange the passes and failures. Let's assume the pass rates for the three parts are P1, P2, and P3, and the failure rates are Q1 = 1 - P1, Q2 = 1 - P2, and Q3 = 1 - P3. The probability can be calculated as:

P = (P1 * P2 * P3) * (Q1 * Q2 * Q3) * (Q1 * Q2 * P3) * (Q1 * P2 * Q3) * (P1 * Q2 * Q3)