Question \( 7 \mid 4 \) marks] The job of a chef culinary artist involves more than simply cooking food; it is about crafting an entire sensory experience where the five senses in human perception is in use. If a chef culinary artist's most used sense is Sight, the probability of that chef's clients' dining experience being of high quality is \( 71 \% \). If a chef culinary artist's most used sense is Hearing, the probability of the dining experience being of high quality is 23\%. If the most used sense is Smell, the probability of the dining experience being of high quality is \( 43 \% \); however, if the most used sense is Taste, the probability of the high-quality dining experience is \( 89 \% \). If the most used sense of the chef culinary artist is Touch, the likelihood of the high-quality dining experience is only \( 17 \% \). Of a batch of chef culinary artists, a survey revealed that the most used sense was Sight for \( 17 \% \) of them, Hearing for \( 2 \% \) of them, Smell for \( 31 \% \), Taste for \( 39 \% \), and Touch for \( 11 \% \) of them. i. A chef culinary artist is selected from the batch to manage the dining experience of some clients. Given that the chef's clients' dining experience is of high quality, what is the probability that the most used sense of the chef culinary artist was Sight or Taste?

To find the probability that the most used sense of the chef culinary artist was Sight or Taste given that the dining experience is of high quality, we can use Bayes' theorem and the law of total probability. Let: - \( S \): the event that the most used sense is Sight - \( T \): the event that the most used sense is Taste - \( H \): the event that the dining experience is of high quality We need to find \( P(S \cup T \mid H) \), which can be calculated using the formula: \[ P(S \cup T \mid H) = P(S \mid H) + P(T \mid H) \] Using Bayes' theorem, we can express \( P(S \mid H) \) and \( P(T \mid H) \) as follows: \[ P(S \mid H) = \frac{P(H \mid S) P(S)}{P(H)} \] \[ P(T \mid H) = \frac{P(H \mid T) P(T)}{P(H)} \] ### Step 1: Calculate \( P(H) \) We first need to calculate \( P(H) \), the total probability of a high-quality dining experience. This can be calculated using the law of total probability: \[ P(H) = P(H \mid S) P(S) + P(H \mid H) P(H) + P(H \mid Sm) P(Sm) + P(H \mid T) P(T) + P(H \mid To) P(To) \] Where: - \( P(H \mid S) = 0.71 \) (probability of high quality given Sight) - \( P(H \mid H) = 0.23 \) (probability of high quality given Hearing) - \( P(H \mid Sm) = 0.43 \) (probability of high quality given Smell) - \( P(H \mid T) = 0.89 \) (probability of high quality given Taste) - \( P(H \mid To) = 0.17 \) (probability of high quality given Touch) And the probabilities of each sense being the most used are: - \( P(S) = 0.17 \) - \( P(H) = 0.02 \) - \( P(Sm) = 0.31 \) - \( P(T) = 0.39 \) - \( P(To) = 0.11 \) Now we can calculate \( P(H) \): \[ P(H) = (0.71 \times 0.17) + (0.23 \times 0.02) + (0.43 \times 0.31) + (0.89 \times 0.39) + (0.17 \times 0.11) \] Calculating each term: 1. \( 0.71 \times 0.17 = 0.1207 \) 2. \( 0.23 \times 0.02 = 0.0046 \) 3. \( 0.43 \times 0.31 = 0.1333 \) 4. \( 0.89 \times 0.39 = 0.3471 \) 5. \( 0.17 \times 0.11 = 0.0187 \) Now summing these values: \[ P(H) = 0.1207 + 0.0046 + 0.1333 + 0.3471 + 0.0187 = 0.6244 \] ### Step 2: Calculate \( P(S \mid H) \) and \( P(T \mid H) \) Now we can calculate \( P(S \mid H) \) and \( P(T \mid H) \): \[ P(S \mid H) = \frac{P(H \mid S) P(S)}{P(H)} = \frac{0.71 \times 0.17}{0.6244} = \frac{0.1207}{0.6244} \approx 0.193 \] \[ P(T \mid H) = \frac{P(H \mid T) P(T)}{P(H)} = \frac{0.89 \times 0.39}{0.6244} = \frac{0.3471}{0.6244} \approx 0.556 \] ### Step 3: Calculate \( P(S \cup T \mid H) \) Now we can find \( P(S \cup T \mid H) \): \[ P(S \cup T \mid H) = P(S \mid H) + P(T \mid H) \approx 0.193 + 0.556 \approx 0.749 \] ### Final Answer Thus, the probability that the most used sense of the chef culinary artist was Sight or Taste given that the dining experience is of high quality is approximately \( 0.749 \) or \( 74.9\% \).