Four Year 12 students want to organise a graduation party - HSC - SSCE Mathematics Advanced - Question 31 - 2023 - Paper 1

To find the probability that Kim is available next Saturday, we can utilize the given conditional probabilities.

1. Given: $P(F) = \frac{3}{10}$ $P(S|F) = \frac{1}{3}$ $P(F|S) = \frac{1}{8}$

2. Using the relationship: $P(S) = P(S|F) \times P(F) + P(S|F') \times P(F')$ where (F') is the event that Kim is not available next Friday. Thus: $P(F') = 1 - P(F) = 1 - \frac{3}{10} = \frac{7}{10}$

3. Now substituting into the previous equation: $P(S) = \frac{1}{3} \times \frac{3}{10} + P(S|F') \times \frac{7}{10}$

4. We know that: $P(S|F') = 1 - P(S|F) = 1 - \frac{1}{3} = \frac{2}{3}$ so, $P(S) = \frac{1}{3} \times \frac{3}{10} + \frac{2}{3} \times \frac{7}{10}$

5. Calculating: $= \frac{1}{10} + \frac{14}{30} = \frac{1 + 14}{30} = \frac{15}{30} = \frac{4}{5}$

To find the probability that at least one of the four students is NOT available next Saturday, it is easier to first compute the probability that all four students are available, and then subtract that value from 1.

1. The probability that Kim is available next Saturday is: $P(S) = \frac{4}{5}$

2. The probability that Kim is NOT available next Saturday is: $P(S') = 1 - P(S) = 1 - \frac{4}{5} = \frac{1}{5}$

3. Since all four students have the same probability of availability, the probability that all four students are available next Saturday is: $P(all \ available) = \left(P(S)\right)^4 = \left(\frac{4}{5}\right)^4 = \frac{256}{625}$

4. Consequently, the probability that at least one student is NOT available is: $P(at \ least \ one \ not \ available) = 1 - P(all \ available) = 1 - \frac{256}{625} = \frac{369}{625} = 0.5904$